It is well-known (and easily shown) that a local quasi-geodesic (for some value of "local") in a $\delta$-hyperbolic space is global (one can compute the constants, as well, from local data). This is false in CAT(0) (think of Euclidean space), but there must be some condition under which it is true. A quick virtual leafing through Bridson-Haefliger brought no joy. (the hidden agenda is figuring out when ping-pong type free subgroups are quasi-isometrically embedded, and with what distortion). Any pointers?
Local-to-global results for "Morse quasigeodesics" in symmetric spaces are proven in section 7 of this paper that I wrote with Bernhard Leeb and Joan Porti.