In this work, we consider the problem of prescribed time scale robot navigation in dynamic environments. Initially, we treat the problem for a special class of configuration spaces, namely sphere worlds, proposing a time-varying control scheme that drives the robot from (almost) all initial configurations to an arbitrary neighborhood of any desired configuration within a predetermined time span, and at the same time prevents any collisions with static and moving obstacles as well as with the workspace boundary along the way. The introduction of a novel vector field allows us to establish the safety of the system and simultaneously apply the Prescribed Performance Control technique to guarantee any predefined transient behavior. Subsequently, we leverage well-established transformations to apply the proposed scheme to the far more practical class of generalized sphere worlds. Finally, we validate the theoretical findings via a non-trivial numerical simulation.