Quantized systems whose underlying classical dynamics possess an elaborate mixture of regular and chaotic motion can exhibit rather subtle long-time quantum transport phenomena. In a short wavelength regime where semiclassical theories are most relevant, such transport phenomena, being quintessentially interference based, are difficult to understand with the system's specific long-time classical dynamics. Fortunately, semiclassical methods applied to wave packet propagation can provide a natural approach to understanding the connections, even though they are known to break down progressively as time increases. This is due to the fact that some long-time transport properties can be deduced from intermediate-time behavior. Thus, these methods need only retain validity and be carried out on much shorter time scales than the transport phenomena themselves in order to be valuable. The initial value representation of the semiclassical propagator of Herman and Kluk [Chem. Phys. 91, 27 (1984)] is heavily used in a number of molecular and atomic physics contexts, and is of interest here. It is known to be increasingly challenging to implement as the underlying classical chaos strengthens, and we ask whether it is possible to implement it well enough to extract the kind of intermediate-time information that reflects wave packet localization at long times. Using a system of two coupled quartic oscillators, we focus on the localizing effects of transport barriers formed by stable and unstable manifolds in the chaotic sea and show that these effects can be captured with the Herman-Kluk propagator.