The status of Jewish ethnicity of descendants of renegade or apostate Jews:[1]

One who is Jewish, remains Jewish even if he or his direct maternal ancestors converted r”l to a different religion. [Although there exist Poskim[2] who challenge or qualify this ruling, practically, the ruling stands as above, and such a Jew’s Kiddushin is considered valid.] Nonetheless, if an apostate Jew [or a descendent of an apostate Jew] desires to re-enter the Jewish community, they are Rabbinically required to immerse in a Mikveh in front of a Beis Din, as is similarly done during the conversion process.[3] [The above applies only if one knows for certain that all of his maternal ancestors were Jewish, born to Jewish mothers. If, however, there is question in this matter, then a full conversion is necessary.[4]]

[1] Michaber and Rama E.H. 44:9 regarding Kiddushin “A Yisrael Mumar who did Kiddushin, it is valid. And even if the Mumar had a child later from a Jewish woman, his Kiddushin is valid” [They do not differentiate between if converted out of duress or willingly]; Rama Y.D. 159:3 [regarding Ribis]; Beis Yosef E.H. 157:6 “The children and all descendants of a Jewish mother who became a Mumar [remains Jewish and their Kiddushin is valid] even after many generations. This matter is an obvious Jewish law and those students who wrote differently, should have been persecuted by the Sages  and excommunicated , as certainly these matters have no root and branch to rely on, and woe top one who is lenient in this” Maharik Shoresh 85; Maggid Mishneh Ishus 4, brought in Beis Yosef 44; Teshuvas Reim 47 and 49 [Rav Eliyahu Mizrachi] that this applies for all the future descendents, see there in length; Noda Beyehuda Tinyana 162, brought in Pischeiy Teshuvah 44:9 that a Mumar remains Jewish; Teshuvas Hageonim Shaareiy Tzedek 3 1:50 [regarding Mumar]; Meiri in Beis Habechira Avoda Zara 26 [regarding Mumar, however, regaridng descendents-see below]; Rashbash 89; 368 [all generations]; Yachin Boaz 2:31 [all generations]; Igros Moshe E.H. 4:83 “It is not possible for a Jew to become a non-Jew. The Muamrim and their children are complete Jews and one cannot join the opposing opinions even as a Safek. The Mahrashdam made a grave error by writing otherwise”; See in great length [50 chapters] Divrei Yatziv Even Haezer 62; Radbaz 3:415; 4:12

Moranos: The descendants of Moreno Jews are considered to have converted out of duress [even though not all were threatened with their lives] and are considered Jewish according to all, so long as they did not intermarry with gentile women. [Teshvas Ream ibid; Rashbash 89; 368; Yachin Boaz 2:31]

[2] Some Poskim rule that a Mumar is considered a gentile and his Kiddushin is invalid. [Opinion in Tur E.H. 44; Ittur Os Kuf 78a “One who desecrates Shabbos in public is like a gentile and his Kiddushin is invalid”; Rav Yehudaiy brought in Beis Yosef 157; Halachos Ketanos 2:240; See Teshuvas Riem ibid for a lengthy discussion on this subject; See Radbaz 3:415 for different sides of this subject; See Aruch Hashulchan E.H. 44:11; See Divrei Yatziv Even Haezer 62 who defends this approach] The Poskim ibid severely negate this approach and it cannot be used in Halacha for any leniency. [Beis Yosef ibid; Igros Moshe ibid]

The descendants of Mumarim: Some Poskim rule that although a Mumar and his/her descendants are Jewish, this only applies if the Mumar converted out of force. If, however, he/she converted out of his/her free will, then all the children born after the conversion are complete gentiles [even if both parents are Jewish]. [Teshuvas Reim ibid in name of Mahariy Chaviv; Rashdam E.H. 10; Kneses Hagedola 44 that from the grandchildren and onwards, this applies even according to Teshuvas Hareim; Beir Heiytiv E.H. 44:8; See Maharibal 2:45; Radab 6; Meiri in Beis Habechira Avoda Zara 26 rules that all the descendents of the Muamr are not Jewish] However, other Poskim vehemently negate this position, saying that a Jew remains a Jew even if his direct maternal ancestor willingly converted many generations ago. [Implication of Michaber ibid; Beis Yosef ibid severely negates the above position; Rashbash ibid; Aruch Hashulchan E.H. 44:11 “There is no root for the above words of the Poskim ibid who negate the Judaism of the descendants, as how exactly does their Jewish identity disappear? So is explained in the Rambam and all Poskim, and so is Setimas Hamichaber in 44:9. The fact that the Rashdam ibid learnt his position from the Talmudic opinion revoking Jewish identity from the ten tribes has no basis, as there it is learned from a verse” Igros Moshe E.H. 4:83 negates the above opinions from Halacha, saying they cannot be joined to any calculation of leniency]

[3] Rama 268:12; 267:8; Shach 267:15; Rabbeinu Simcha; Radbaz 3:415; See however Rashbash 68, brought in Pischeiy Teshuvah 269:10, that the Moranos do not need to immerse at all

[4] Radbaz 3:415