In terms of responsibilities, it depends on the Rabbinic position. Technically a Rabbi is anyone who received certifying authority to deliberate on Jewish Law. In practice, it's also extended to anyone with a significant degree of subjectively greater knowledge, even without authority to deliberate on Jewish Law. I'm pretty sure one of the major leading Israeli Haredi Rabbis has never received that authority, but he's memorized word-for-word perhaps hundreds of Jewish texts.
So it really depends on the particular position. There can be a communal Rabbi who functions similar to a Grand Rabbi, in that he guides his community, even if they aren't part of his congregation (or he doesn't have one).
There's a pulpit Rabbi who may either guide his particular congregation, or simply be an available authority on Jewish Law and perhaps give a sermon.
So in terms of Hassidism, you may have the Grand Rabbi of a particular Hassidic sect who's the leader of the sect. He may institute enactments for his community, give sermons, etc. However, since his community may extend beyond the town he lives in, other congregations of his sect may hire (or he may be asked to appoint) a different Rabbi as a local authority and preacher.
Since the Grand Rabbi isn't a hired pulpit Rabbi, but a community Rabbi, you don't really vote him out. Instead he relies on people choosing to follow him (as do all community Rabbis). So anyone can change their "allegiance" and switch to a different Hassidic sect or choose to follow a different Grand Rabbi. I don't think I've ever heard of a Grand Rabbi ever being tossed to the curb, but it's definitely not uncommon for individuals to switch to other Rabbis/sects or even entire groups of people to choose to follow a different Grand Rabbi.