It is a well known fact that velocity grandients in a flow change the dispersion of a passive tracer. One clear manifestation of this phenomenon is that in systems with homogenization type diffusive long time/large scale behavior, the effective diffusivity often differs greatly from the molecular one. An important aspect of these well known result is that they are only valid on timescales much longer than the inverse diffusivity. We are interested in what happens on shorter timescales (subhomogenization regimes) in a family of two-dimensional incompressible periodic flows that consists only of pockets of recirculations essentially acting as traps and infinite flowlines separating these where significant transport is possible. Our approach is to follow the random motion of a tracer particle and show that under certain scaling it resembles time-changed Brownian motions. This shows that while the trajectories are still diffusive, the variance grows differently than linear.