I thought one has to distinguish between an object-based prohibition that will become permitted later and person-based prohibitions that will become permitted later. For example, leavened food on Passover or untithed produce are object-based prohibitions that will become permitted later, but the dairy food here does not involve an object-based prohibition, only a person-based prohibition, and therefore the rule of “instead of eating it in prohibition, eat it in permission” would not apply here. By the way,

I saw a source on WikiYeshiva regarding an oath about a food item (= a person-based prohibition) that says this:
An oath — for example, if one swore not to eat a certain thing — medieval authorities wrote that it is considered something that will become permitted later (this seems implied by Maimonides, Vows 5:14-15; Ritva on Shevuot 4a; this also seems implied by Migdal Oz there 15, and Radbaz there 16; Pri Megadim, Introduction to Passover 2:1:16), because it is a commandment to seek release from it (Ritva there); but some later authorities wrote that an oath is not considered something that will become permitted later, because there is no commandment to seek release from the oath, and on the contrary, one seeks release from an oath only under pressure (see the entry “oath”); and if someone swore not to eat a certain item and that item became mixed with others, it is nullified by majority (Beit Yaakov 110; Teshuvah Me’Ahavah 1:102, 104; Pitchei Teshuvah, Yoreh De’ah 102:10, in their name).

That is, even according to some of the later authorities, an oath (and perhaps also all person-based prohibitions) is not considered something that will become permitted later.

And similarly I found on WikiYeshiva regarding naziriteship, that it is not considered something that will become permitted later. For example, if grape wine became mixed with fig wine, and the nazir wants to drink it: even though he could go to a sage to seek annulment, it is not considered something that will become permitted later, because the inquiry to the sage concerns the naziriteship itself (that he should cancel the naziriteship) and not the prohibited items themselves (such as drinking the wine), and only when the sage uproots the naziriteship does the prohibition on drinking wine lapse automatically; therefore it is not considered something that will become permitted later (Tzafnat Pa’neach, Klalei HaTorah VehaMitzvot, vol. 1, s.v. “something that will become permitted later” in naziriteship).

What do you think?

The prohibition of meat and milk is an object-based prohibition. However, the obligation to wait between meat and dairy is rabbinic and perhaps applies only to the person (though I’m not sure about that either).
By the way, with naziriteship it seems to me that the reasoning is a bit different. There we are talking about the permission, not the prohibition. The permission applies not to the item but to the person.
As for an oath, that reasoning seems puzzling to me. Fine, if you say it’s a person-based prohibition. But the fact that there is no commandment to seek release — there is also no commandment to seek release from a vow, and it is explicit in tractate Nedarim that a vow is considered something that will become permitted later.

Moreover, the reasoning for distinguishing between person-based and object-based prohibitions seems to me more relevant for nullification. But regarding the laws of doubt, I do not see any distinction. Instead of eating it in prohibition, eat it in permission.

You wrote regarding naziriteship that the permission applies not to the item but to the person. That is exactly what I was trying to say: in person-based prohibitions, the permission applies not to the thing (the object) but to the person. And therefore, in person-based prohibitions, the rule of something that will become permitted later does not apply (as with naziriteship), and so I am trying to argue that doubts in a rabbinic person-based prohibition that will become permitted later are treated like any other rabbinic doubt, where we rule leniently (as opposed to a doubt in object-based prohibitions that will become permitted later). As for the dairy food, there is no prohibition in the food itself so long as it was not mixed with meat; only after you eat it does the prohibition take effect in reality (say, when it mixes in the stomach), and therefore in a case of doubt we should be lenient (because this is a rabbinic rule).

As I added, regarding the rule that in a rabbinic-level doubt we rule leniently, I really do not understand the reasoning for distinguishing between person and object at all. Maybe regarding nullification one could say that.
What I meant to say regarding naziriteship is that you raised two formulations: is the prohibition on the thing or on the person? Is the permission on the thing or on the person? Thus, for example, with a vow the prohibition is on the thing, but if there were permission for the person, that would not render the thing itself permitted. But that is only a side remark.
And regarding eating dairy food, that is exactly what I said. If it mixes in the stomach, or alternatively if the milk becomes an object-based prohibition when you have eaten meat, then this too is an object-based prohibition.

The reasoning for the distinction, in my opinion, comes from the Talmudic source for the rule of something that will become permitted later. They derive this rule from an egg laid on a Jewish holiday. That is, I am trying to argue that this whole rule applies only to prohibited objects like an egg laid on a holiday, leavened food on Passover, untithed produce, etc., but does not apply to prohibitions that are not in the object itself, such as wine for a nazir, or just food on a fast day. For example, if a person wants to eat at twilight at the end of the Ninth of Av, seemingly there is a possible rabbinic prohibition here (= leniently), but on the other hand it is a prohibition that will become permitted later (once the stars come out it will be permitted), and still I am trying to argue that it would be permitted to be lenient in that prohibition without waiting for nightfall, because all the laws of “something that will become permitted later” apply only to prohibited objects, not to person-based prohibitions such as the obligation to fast on the Ninth of Av.

Sometimes halakhic formalism does indeed work that way. But the reasoning behind something that will become permitted later applies both to object-based prohibitions and to person-based prohibitions: instead of permitting him to eat something that is possibly prohibited, let him wait a bit and eat it with full permission.

If a person ate meat and now, one hour after eating the meat, wants to eat something pareve that was cooked in a dairy pot — what is the law? And the reason?

http://www.yeshiva.org.il/midrash/19282

Following up on this question: the rule that one should be stringent in doubts involving something that will become permitted later — does that apply only to doubts about the facts, or also to doubts in the law? For example, there is a dispute among halakhic decisors whether one must wait some amount of time after eating cheese before eating meat is permitted. Would it be correct to follow the rule that in a rabbinic-level doubt we rule leniently, or should one be stringent דווקא because of the rule that a doubt involving something that will become permitted later is treated stringently?

Seemingly, in a legal doubt there is reason to be stringent even without invoking something that will become permitted later, at least in a dispute among authorities, because here perhaps both sides are correct (“these and those are the words of the living God”). In any case, at least regarding something that will become permitted later, it seems obvious to me that one should be stringent in that case too, for the same reason: instead of eating it in prohibition, eat it in permission.

If I understand you correctly, in every rabbinic legal doubt one should be stringent? But don’t the halakhic decisors generally use the common rule that in a rabbinic-level doubt we rule leniently even regarding halakhic disputes in rabbinic laws?

No. I wrote that there is reason for such a view. For example, regarding a double doubt, some have written that in a legal doubt there is no permission to be lenient. See, for example, here:
http://www.yeshiva.org.il/wiki/index.php?title=%D7%A1%D7%A4%D7%99%D7%A7%D7%90_%D7%93%D7%93%D7%99%D7%A0%D7%90

Actually, from the source you brought I understood the opposite. It says there:
“In a rabbinic-level doubt, one is lenient even in a dispute among halakhic decisors (Mishnah Berurah 160:47).”
And in addition it says:
“Double doubt — we do not say double doubt in a dispute among halakhic decisors (Shev Shema’teta 1:23 in the name of the Mahari ben Lev).”
That is, I think they mean that there is no difference between a double doubt and a regular doubt in a dispute among halakhic decisors. And from here it follows that a legal doubt in a Torah-level matter is treated stringently, and even a double doubt in a Torah-level legal matter is treated stringently (there is no difference between one doubt and a double doubt). And the same also in rabbinic law: let us say there is a dispute among halakhic decisors regarding law A. And even among the camp of decisors who are lenient regarding law A, there is a dispute among them whether regarding law B, which branches off from it, one should be lenient or stringent. If both laws A and B are rabbinic, then even though there is a double doubt regarding law B whether one may be lenient about it — the first doubt is perhaps the stringent camp regarding A is correct (and then one should be stringent also regarding B), and the second doubt is that even if we say the lenient camp regarding A is correct, perhaps the stringent camp regarding B is correct — still one should be lenient regarding law B because of the rule that in a rabbinic-level doubt we rule leniently. Did I understand correctly?

By the way, I noticed that something else is written there contrary to what you said above regarding my original question:
“Something that will become permitted later does not prohibit in a legal doubt, only in a factual doubt (Rabbi Akiva Eiger, Berakhot 2a, end of s.v. ‘There’ and ‘From where,’ brings the Pri Chadash regarding a rabbinic-level doubt, and adds on his own also regarding double doubt).”

Indeed. In a rabbinic-level doubt one is lenient even in a dispute among halakhic decisors, and that is what I wrote. And regarding double doubt, we are not talking about two doubts among the decisors but a doubt built on top of a doubt among the decisors. In a Torah-level matter they are stringent in that case because the first doubt is not considered a real doubt (both of its sides are true), and that is what I wrote.

As for something that will become permitted later, I do not remember that, but perhaps Rabbi Akiva Eiger does write it. From reasoning alone, it seems to me not so. Though perhaps the same reasoning applies here too: in a dispute among halakhic decisors both sides are valid, and therefore one can rely on the lenient view, and it is not considered that you are now eating in prohibition (which is the reasoning for not eating and postponing until one may eat in full permission).