  This paper studies wormhole solutions to Einstein gravity with an arbitrary
number of time dependent compact dimensions and a matter-vacuum boundary. A new
gauge is utilized which is particularly suited for studies of the wormhole
throat. The solutions possess arbitrary functions which allow for the
description of infinitely many wormhole systems of this type and, at the
stellar boundary, the matter field is smoothly joined to vacuum. It turns out
that the classical vacuum structure differs considerably from the four
dimensional theory and is therefore studied in detail. The presence of the
vacuum-matter boundary and extra dimensions places interesting restrictions on
the wormhole. For example, in the static case, the radial size of a weak energy
condition (WEC) respecting throat is restricted by the extra dimensions. There
is a critical dimension, D=5, where this restriction is eliminated. In the time
dependent case, one \emph{cannot} respect the WEC at the throat as the time
dependence actually tends the solution towards WEC violation. This differs
considerably from the static case and the four dimensional case.
