We consider spherically symmetric accretion of material from an initially homogeneous, uniformly expanding medium onto a Newtonian point mass M. The gas is assumed to evolve adiabatically with a constant adiabatic index F, which we vary over the range Γ ∊ [1, 5/3]. We use a one-dimensional Lagrangian code to follow the spherical infall of material as a function of time. Outflowing shells gravitationally bound to the point mass fall back, giving rise to a inflow rate that, after a rapid rise, declines as a power law in time.If there were no outflow initially, Bondi accretion would result, with a characteristic accretion time-scale ta,0. For gas initially expanding at a uniform rate, with a radial velocity U = R/t0 at radius R, the behavior of the flow at all subsequent times is determined by ta,0/t0. If ta,0/t0 ≫ 1, the gas has no time to respond to pressure forces, so the fluid motion is nearly collisionless. In this case, only loosely bound shells are influenced by pressure gradients and are pushed outward. The late-time evolution of the mass accretion rate Mdot is close to the result for pure dust, and we develop a semianalytic model that accurately accounts for the small effect of pressure gradients in this limit. In the opposite regime, ta,0/t0 ≪ 1, pressure forces significantly affect the motion of the gas. At sufficiently early times, t ≤ ttr, the flow evolved along a sequence of quasi-stationary, Bondi-like states, with a time-dependent Mdot determined by the slowly varying gas density at large distances. However, at later times, t ≥ ttr, the fluid flow enters a dustllke regime; ttr is the time when the instantaneous Bondi accretion radius reaches the marginally bound radius. The transition time ttr depends sensitively on ta,0/t0 for a given Γ and can greatly exceed t0. We show that there exists a critical value Γ = 11/9, below which the transition from fluid to ballistic motion disappears. As one application of our calculations, we consider the fallback of initally outflowing gas onto the compact remnant in the core of a Type II supernova. The results have important implications for determining whether the remnant in SN 1987A is a neutron star or a black hole. We demonstrate that the outcome of fallback depends sensitively on initial conditions, principally on the sound speed of the material at the onset of infall. If the sound speed is small initially, Cs ≤ 300-400 km s-1, then the mass accretion rate remains super-Eddington for many years after the explosion, and the total mass accreted is substantial, perhaps enough to drive collapse of the neutron star to a black hole for a sufficiently soft equation of state. On the other hand, if the sound speed is considerably larger at the onset of infall, Cs ∼ 104 km s-1 or so, both the mass accretion rate and the total mass accreted may be small enough that a neutron star could lie at the core of SN 1987A.