  Motion under stochastic resetting serves to model a myriad of processes in
physics and beyond, but in most cases studied to date resetting to the origin
was assumed to take zero time or a time decoupled from the spatial position at
the resetting moment. However, in our world, getting from one place to another
always takes time and places that are further away take more time to be
reached. We thus set off to extend the theory of stochastic resetting such that
it would account for this inherent spatio-temporal coupling. We consider a
particle that starts at the origin and follows a certain law of stochastic
motion until it is interrupted at some random time. The particle then returns
to the origin via a prescribed protocol. We study this model and surprisingly
discover that the shape of the steady-state distribution which governs the
stochastic motion phase does not depend on the return protocol. This shape
invariance then gives rise to a simple, and generic, recipe for the computation
of the full steady-state distribution. Several case studies are analyzed and a
class of processes whose steady-state is completely invariant with respect to
the speed of return is highlighted. For processes in this class we recover the
same steady-state obtained for resetting with instantaneous
returns---irrespective of whether the actual return speed is high or low. Our
work significantly extends previous results on motion with stochastic resetting
and is expected to find various applications in statistical, chemical, and
biological physics.
