The institution Hasidism invented is the court. A rebbe is not primarily a legal authority, and in many courts the halakhic questions go to a rabbi who is somebody else. What the rebbe offers is a channel: the doctrine of the tsaddik holds that a person of sufficient attachment to God draws blessing down to those bound to him, and that a follower reaches higher through that bond than alone.
The forms this takes are specific and have barely changed. A follower writes a kvitl, a slip of paper with a name, a mother name and a request, and hands it over at a private audience. The tish is a table, at which the rebbe eats on the sabbath in front of hundreds of men who watch, sing between courses, and receive shirayim, remnants of the food he has touched. Pilgrimage to the court for the high holidays or for a yahrzeit gathers followers who live elsewhere.
Succession changed the movement more than any doctrine did. The earliest leaders were recognised for what they were; within two generations the position passed from father to son, and the courts became dynasties named for the towns they had grown in. The names outlived the towns entirely. Ger, Satmar, Belz, Vizhnitz and Bobov are places in Poland, Hungary, Ukraine and Galicia, and the communities that carry those names now live in Brooklyn, Jerusalem, Bnei Brak, London and Antwerp.