Gemara

(Ula, citing R. Yochanan): The daughter of a converted Amoni may marry a Kohen.

Question (Rava bar Ula): Like whom is this?

It is unlike R. Yehudah, for he says that a convert's daughter is like a Chalal's daughter!

Answer #1: It is like R. Yosi.

Objection: If so, the law is obvious! R. Yosi permits even the daughter of two converts to a Kohen!

Answer: One might have thought that that he permits the daughter of a convert who may marry Yisraelim, but not of a forbidden convert.

Kidushin 77a (Mishnah - R. Yehudah): The daughter of a male convert is (forbidden to Kehunah) like the daughter of a Chalal.

R. Eliezer ben Yakov says, if a Yisrael married a convert, or a convert married a Bas Yisrael, the daughter is permitted to Kehunah. The daughter of two converts is forbidden to Kehunah.

The law of converts and freed slaves is the same. Even after 10 generations, the daughter is forbidden to Kehunah unless her mother is a Bas Yisrael.

R. Yosi says, even the daughter of two converts is permitted to Kehunah.

78a (Beraisa - R. Yehudah): We learn from a Kal va'Chomer that a male convert's daughter is like a Chalal's daughter:

A Chalal was fathered by a Yisrael, yet his daughter is disqualified (from Kehunah). A convert was fathered by a Nochri, all the more so his daughter is disqualified!

Question: We cannot learn from a Chalal, for he (or one of his ancestors) was conceived through sin!

Answer: We learn from a Mitzri Rishon (he himself converted). Even though he was not formed through sin, his daughter is forbidden.

Question: We cannot learn from a Mitzri Rishon, for he may not marry a Bas Yisrael!

Answer: We learn from a Tzad ha'Shavah of a Chalal and a Mitzri Rishon. Both are different from the Tzibur (one was formed through sin and one cannot marry a Bas Yisrael), and their daughters are disqualified. Also a convert is different from the Tzibur (he was fathered by a Nochri), so his daughter is disqualified!

Objection: We cannot learn from a Chalal and a Mitzri. Both of them disqualify a woman (from Kehunah) by having Bi'ah with her!

Answer: Indeed, R. Yehudah learns that a convert also disqualifies a woman by having Bi'ah with her, from this Tzad ha'Shavah!

(Beraisa - R. Shimon: If a girl converted before three years old, she is permitted to Kehunah. Pinchas (a Kohen) was among the soldiers who took captives from Midyan, and they were told "Keep all the young girls alive for yourselves (they are permitted to you)";

Chachamim say, "keep them alive" for slaves.

The different opinions among the Tana'im were learned from "They (Kohanim) will not marry a widow or divorcee, rather virgins from the seed of Yisrael":

R. Yehudah expounds that all the (primary) seed (i.e. the father's) must be from Yisrael;

R. Eliezer ben Yakov expounds "from the seed of Yisrael" - it suffices if one parent is a Yisrael;

R. Yosi expounds that she must be seeded in Yisrael (her parents are Yisraelim);

R. Shimon expounds that she must be a Yisraelis when her Besulim (tokens of virginity) finish growing (at the age of three).

(Mishnah - R. Yosi): Even if a convert married a convert (his daughter is permitted to Kehunah).

(Rav Hamnuna): The Halachah follows R. Yosi.

(Rabah bar bar Chanah): The Halachah follows R. Yosi, but after the Churban, Kohanim conduct stringently like R. Eliezer ben Yakov.

(Rav Nachman): If a Kohen asks, we tell him that the Halachah follows R. Eliezer ben Yakov. If he married the daughter of converts, he may stay married, like R. Yosi.

Rishonim

Rif and Rosh (Kidushin 32a and 4:15): The Halachah follows R. Yosi, but after the Churban, Kohanim conduct stringently like R. Eliezer ben Yakov. If a Kohen asks, we tell him that the Halachah follows R. Eliezer ben Yakov. If he married the daughter of converts, he may stay married, like R. Yosi.

Rambam (Hilchos Isurei Bi'ah 19:12): If converts and freed slaves married among themselves, even after many generations the daughter is forbidden to a Kohen. If she married she need not leave, because she was conceived and born in Kedushah (i.e. from Yisraelim).

Beis Yosef (EH 7 DH Ger she'Nasa): Obviously, she stays only if she was conceived and born in Kedushah, for a convert less than three years old is Pasul to Kehunah!

Question (Chelkas Mechokek 29): It is not obvious that if she was conceived Lo bi'Kdushah but was born in Kedushah that she must leave!

Answer (Beis Shmuel 42): The Halachah does not follow R. Shimon; a convert even less than three is forbidden. Since we require one aspect (birth) in Kedushah, it is reasonable that also the other (conception) must be.

Beis Moshe (36, in Otzar ha'Poskim): If a woman converted before birth, b'Di'eved her daughter may stay married to a Kohen, like R. Eliezer ben Yakov!

Question (Yad Aharon, Hagahos Beis Yosef): Why must the Beis Yosef bring a roundabout proof? R. Yosi requires 'Nizre'u b'Yisrael'!

Rambam (ibid.): If a male or female convert or freed slave married a Yisraelis or a Yisrael, the daughter is permitted to a Kohen l'Chatchilah.

Poskim

Shulchan Aruch (EH 7:21): If two converts married, their daughter is forbidden to a Kohen. Also the daughter's daughter is forbidden, even after many generations. If she married a Kohen she need not leave. If one of them parents is a Yisrael, the daughter is permitted to a Kohen l'Chatchilah.