The foundational rule, stated by the Shulchan Arukh (OC 53:25), is that a chazzan may not be removed from his position unless a specific disqualification is found in him — and the Rema adds that even substantiated communal gossip alone does not suffice without witnesses.
The Mishnah Berurah (53:15) clarifies that once a chazzan has established a chazakah in his position, the bar for removal is high, and even if a persistent rumor arises only a single objector may press for his removal — but where he has done teshuvah for a past sin, all agree he is not removed.
The Magen Avraham (53:21) rules that it is not proper to displace an established chazzan merely because another candidate has a more pleasing voice, and that if the majority wishes the new candidate, both should serve together rather than ousting the incumbent.
The question of chazakah in a mitzvah more broadly is addressed by the Shulchan Arukh (OC 153:22), which holds that a person accustomed to a communal mitzvah retains his claim to it so long as he did not voluntarily relinquish it — a principle that informs how chazakah over the amud is analyzed when circumstances change.
The Arukh HaShulchan (OC 53:21) notes that in practice the role of the Yamim Noraim baal tefillah has become the central and defining function of the communal chazzan, and that a minority of the congregation cannot prevent the majority from engaging the cantor of their choice — though equally, an established chazzan's position is not lightly overturned by a minority's objection.
Igros Moshe (OC 3:13) addresses precisely the merger scenario: a baal tefillah who held a chazakah in one synagogue prior to its union with a second congregation, where members of the second shul object to retaining him on account of his weak voice — and, on our reading of the cited page, Rabbi Feinstein analyzes the halachic limits of that chazakah and the conditions under which replacement on such grounds may or may not be permitted.
Mishneh Halachos (11:41) addresses a parallel communal dispute and concludes, on our reading of the cited page, that an individual who is a genuine participating member of a synagogue community — sharing its burdens and obligations — retains rights across all of that community's frameworks, a principle that bears directly on the standing of an incumbent baal tefillah within a newly merged kehillah.