Kavod HaRav: Thank you, amazing!! Is there any rule that once a person lives in Jerusalem to fulfill that mitzvah, that they shouldnt move away from Jerusalem?

ב"ה Shalom, Rabbi Eliezer Waldenberg zt"l dealt with this question extensively in his Responsa Tzitz Eliezer Vol. 13: 22. He actually says that this is no simple question, based upon the Mishna in Ketubot 110b, which says that: "All may force their family to ascend to Eretz Yisrael, i.e., one may compel his family and household to immigrate to Eretz Yisrael, but all may not remove others from Eretz Yisrael, as one may not coerce one’s family to leave. Likewise, all may force their family to ascend to Jerusalem, and all may not, i.e., no one may, remove them from Jerusalem". The laws in this Mishna are codified in the Rambam Hilchot Ishut 13:20, and in the Shulchan Aruch Even Haezer 75:4. Rav Waldenberg concludes that if one may force his or her spouse to move to Jerusalem and not to leave Jerusalem for another place in Eretz Yisrael, it follows that the law applies to a single person who has been dwelling in the sanctified Jerusalem must not to leave it even to a nicer neighborhood elsewhere in Eretz Yisrael. Rav Waldenberg then presents two answers from two Rabbis who lived in Jerusalem between 250- 300 years ago, who were both asked the same question, in regard to someone who had been living in Jerusalem, if he may leave the city even for the sake of praying at Me'arat Hamachpela, the tomb of our patriarchs and matriarchs. He quotes, Rabbi Avraham Meyuchas in his book Sdeh Ha'aretz (Vol. III: 11 ) and his Rabbi, Yisrael Meir Halevi in his book, Pri Ha'aretz ((Vol. III: 7 ) who both dealt with this question seriously and both ruled that one may leave Jerusalem temporarily but not permanently. Rav Waldenberg further says, that it seems that this virtue of Jerusalem is brought about the custom of some unique men of distinction who never left Jerusalem even for one night. He then adds, that the prohibition of leaving Jerusalem only applies to those living in the Old City not to those who live in the new neighborhoods of Jerusalem. In another answer, (Vol.14: 52) Rav Waldenberg adds that that when one is doubt where the ancient borders of Jerusalem were, one may be lenient in regard to leaving Jerusalem and in any case most people are lenient on this issue, and as he said in the previous answer, that this was the practice of only a rare breed of men of distinction in each generation. Nonetheless, Rav Waldenberg adds that although the new neighborhoods of Jerusalem do not have the same spiritual value as the ancient part, they still have the virtue of being close to the location of the Bet Hamikdash. There is so much more to be said about Jerusalem, but perhaps I'll just conclude by quoting the 12th century , Rav Yehuda Halevi , who wrote at the end of his book "Sefer Hakuzari" that "Jerusalem shall only be rebuilt when Yisrael yearns for her to the utmost extent that they embrace her stones and earth" All the best