  We study the duration and variability of late time X-ray flares following
gamma-ray bursts (GRBs) observed by the narrow field X-ray telescope (XRT)
aboard the {\it Swift} spacecraft. These flares are thought to be indicative of
late time activity by the central engine that powers the GRB and produced by
means similar to those which produce the prompt emission. We use a
non-parametric procedure to study the overall temporal properties of the flares
and a structure function analysis to look for an evolution of the fundamental
variability time-scale between the prompt and late time emission. We find a
strong correlation in 28 individual x-ray flares in 18 separate GRBs between
the flare duration and their time of peak flux since the GRB trigger. We also
find a qualitative trend of decreasing variability as a function of time since
trigger, with a characteristic minimum variability timescale $\Delta t/t=0.1$
for most flares. We interpret these results as evidence of internal shocks at
collision radii that are larger than those that produced the prompt emission.
Contemporaneous detections of high energy emission by GLAST could be a crucial
test in determining if indeed these X-ray flares originate as internal shocks
behind the afterglow, as any X-ray emission originating from behind the
afterglow is expected to undergo inverse Compton scattering as it passes
through the external shock.
