A finite $W$-algebra $U(\g,e)$ is a certain finitely generated algebra that
can be viewed as the enveloping algebra of the Slodowy slice to the adjoint
orbit of a nilpotent element $e$ of a complex reductive Lie algebra $\g$. It is
possible to give the tensor product of a $U(\g,e)$-module with a finite
dimensional $U(\g)$-module the structure of a $U(\g,e)$-module; we refer to
such tensor products as translations. In this paper, we present a number of
fundamental properties of these translations, which are expected to be of
importance in understanding the representation theory of $U(\g,e)$.