Another nation that had restrictions on converting and marrying in was Edom, descended from Jacob’s brother Esau. Unlike Ammon and Moab, however, these restrictions only apply to the first two generations, no longer. So an Edomite convert could not join the general marriage pool, nor could his son, but his grandson was free and clear. (This does not obligate any particular individual to marry the grandson of an Edomite convert, it merely prohibits excluding them from consideration on the faulty assertion that such a marriage is forbidden.)
The reason for this mitzvah is that God doesn’t want us to assume that Edom necessarily deserves the same treatment as Ammon and Moab, even though they have historically caused us much grief. We have a relationship with Edom that when one of us is up, the other is down (see, for example, Genesis 27:40). That’s just the way things are and we’re not to hold it against our Edomite cousins.
It’s worth noting that, throughout the Talmud, Edom is the name for Rome. It is not clear, however, whether the Romans are the actual descendants of the Biblical nation of Edom or whether they merely inherited the name, perhaps through conquest.
Also worth mentioning is that there’s at least one noteworthy Edomite convert in Tanach: the prophet Ovadya (Obadiah), whose book is part of Trei Asar (the 12 “minor” prophets). A convert from Edom, Ovadya was a servant of King Ahab. He hid 100 prophets from Ahab’s wife Jezebel, who sought to eradicate them (I Kings chapter 18). According to tradition, the widow assisted by the prophet Elisha in 2 Kings chapter 4 was Ovadya’s wife.
This mitzvah applies in all times and places. It is discussed in the Talmud in the tractates of Yevamos (76b, 77b) and Kiddushin (75a) and is codified in the Mishneh Torah in the twelfth chapter of Hilchos Issurei Bi’ah. This mitzvah is #54 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos but it is not listed in the Chofetz Chaim’s Sefer HaMitzvos HaKatzar.