I think I understand that East Berliners could travel freely to West Berlin until the Berlin Wall was erected to stop it, and continued to attempt to escape to West Berlin even after that. But once someone entered West Berlin, how did they then move to the West since they were still landlocked by East Germany? Were they given West German citizenship and flown out?
The usual (and officially encouraged procedure) was to go the refugee center at Berlin-Marienfelde, where you were "checked in" and provided with Western papers, where you would stay until a relocation home (most likely in West Germany) could be found for you. Space in West Berlin was limited, and most refugees wanted to get to West Germany rather than stay in West Berlin.
The (West German) constitution regarded West Germany as the sole legal representative of the German nation proper, which entitled East Germans to immediate Western citizenship.
And yes, once given those papers, you had free passage to West Germany, mostly via air. (Incidentally, one East German plan to cut off the refugee flow was to just get in control of the air passenger control, which would have been via forcing the West to accept to have all West Berlin flights checked by East German or Soviet personnel, preferably on East German soil; this plan was never followed up on, but did make it into East German discussions with the West in early 1961).
Referred to in: Rueger, Fabian, Kennedy, Adenauer and the Making of the Berlin Wall, 1958-1961, Stanford Univ. (Dissertation), 2011.