We define "random trip", a generic mobility model for random, independent node motions, which contains as special cases: the random waypoint on convex or nonconvex domains, random walk on torus, billiards, city section, space graph, intercity and other models. We show that, for this model, a necessary and sufficient condition for a time-stationary regime to exist is that the mean trip duration (sampled at trip endpoints) is finite. When this holds, we show that the distribution of node mobility state converges to the time-stationary distribution, starting from the origin of an arbitrary trip. For the special case of random waypoint, we provide for the first time a proof and a sufficient and necessary condition of the existence of a stationary regime, thus closing a long standing issue. We show that random walk on torus and billiards belong to the random trip class of models, and establish that the time-limit distribution of node location for these two models is uniform, for any initial distribution, even in cases where the speed vector does not have circular symmetry. Using Palm calculus, we establish properties of the time-stationary regime, when the condition for its existence holds. We provide an algorithm to sample the simulation state from a time-stationary distribution at time 0 ("perfect simulation"), without computing geometric constants. For random waypoint on the sphere, random walk on torus and billiards, we show that, in the time-stationary regime, the node location is uniform. Our perfect sampling algorithm is implemented to use with ns-2, and is available to download from http://ica1www.epfl.ch/RandomTrip