We consider the interaction between a marginally stable Bonnor-Ebert (BE) sphere and the surrounding ambient medium. In particular, we explore how the infall from an evolving ambient medium can trigger the collapse of the sphere using three-dimensional adaptive mesh refinement simulations. We find the resulting collapse dynamics to vary considerably with ambient density. In the highest ambient density cases, infalling material drives a strong compression wave into the cloud. It is the propagation of this wave through the cloud interior that triggers the subsequent collapse. For lower ambient densities, we find the main trigger of collapse to be a quasistatic adjustment of the BE sphere to gravitational settling of the ambient gas. In all cases, we find that the classic "outside-in" collapse mode for super-critical BE spheres is recovered before a protostar (i.e., sink particle) forms. Our work supports scenarios in which BE dynamics naturally begins with either a compression wave or infall dominated phase, and only later assumes the usual outside-in collapse behavior.