No. There are trajectories that are not asymptotically safe. But such trajectories may pass near enough to the fixed point that the fixed point properties affect the low energy theory.

http://arxiv.org/abs/1008.3621
"Now let us consider statement (2) from the point of view of “emergence”, i.e. for a trajectory that is close to asymptotic safety but not exactly safe. The closer such trajectory gets to the FP, the longer the time it passes there."