Re Rav Moshe Feinstein zt”l and Lubavitch – it is very important to know that the Agudas Harabbonim, of which Rav Moshe was President in his later years, had a Lubavitcher secretary, by the name of Avrohom Shmuel Lewin. So therefore purported statements by Rav Moshe from that period re Lubavitch need to be taken with at least a grain of salt, keeping that important detail in mind. וד”ל. Rav Moshe was then elderly and weaker physically. In fact, the only time (IIRC) that Rav Moshe was at 770 E.P. and met with the Lubavitch leader, was when he came to Crown Heights for the wedding of the aforementioned Lewin.
Also, it should be noted that Rav Moshe had already been niftar by the time the messianic frenzy of Chabad-Lubavitch peaked in the last years of their late leader’s life. So he did not see certain things. To claim that he gave a blanket endorsement of everything of Lubavitch from his time to eternity, is, of course, absurd.
Additionally, if he met the leader of the Chabad only one time in the circa forty five years when they overlapped in NYC, that itself speaks loudly.