We examine the dynamics of skyrmions in systems with a region of strong pinning coexisting with a pin-free region, where the equilibrium state has a uniform skyrmion density. Under an applied drive, skyrmions accumulate in the pin-free region along the edge of the pinned region due to the skyrmion Hall effect. As the drive increases, a series of dynamical structural transitions occur in the flowing skyrmion lattice similar to those observed in the compression dynamics of crystals. These transitions correspond to reductions in the number of flowing rows of skyrmions due to the collective motion of the skyrmions into the pinned region, and they are accompanied by a series of steps in the velocity force curves leading to a negative differential conductivity. When the number of pinning sites is sufficiently large, a drive-induced pinning effect can occur in which the skyrmion Hall effect forces all of the skyrmions to enter the strongly pinned region. This reentrant pinning effect becomes more pronounced for increasing intrinsic skyrmion Hall angle and larger pinning sites.