Is there any problem with moving to Eretz Yisroel (on Aliyah) during the first part of the Three Weeks?
It is permitted to move to Eretz Yisrael during this time.
Sources: The mitzvah of settling the Land of Israel is a great mitzvah, to the degree that it even defers the rabbinic prohibition of having a document of sale written by a non-Jew. Even in the Nine Days, poskim write that one who needs the space to live may build a house (or move houses), and it would appear that the same applies to somebody moving to Israel. See Orchos Chaim 551:8; Mekor Chaim 551:2.
Based on Biur Halachah 223:3, the blessing one should make when moving in to a new house with one’s family is hatov vehameitiv, and this alleviates the problem of reciting the shehechianu blessing during the Three Weeks.