The Torah commands us to love converts: ואהבתם את הגר (Ekev 10:19)
Are the children of converts considered "converts" in this context, or are they considered normal Jews?
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The Talmud, Sanhedrin 94a, has a discussion about what "VaYichad Yitro" (Shemot 18:10) means. Shmuel says (as Rashi explains) that it means that he was very pained by the news of the downfall of Egypt. Rav (or Rav Pappa) says that this is what the people mean when they say that a convert, even up to 10 generations after the conversion, you should not disparage and Aramean in his presence.
היינו דאמרי אינשי גיורא עד עשרה דרי לא תבזה ארמאי קמיה
This is the popular saying, "A convert, until 10 generations, do not disparage an Aramean in his presence."
So we see that even the descendant of a convert called a convert in this case, and requires extra sensitivity.