  We determine the asymptotic spreading speed of the solutions of a Fisher-KPP
reaction-diffusion equation, starting from compactly supported initial data,
when the diffusion coefficient is a fixed bounded monotone profile that is
shifted at a given forcing speed and satisfies a general uniform ellipticity
condition. Depending on the monotony of the profile, we are able to
characterize this spreading speed as a function of the forcing speed and the
two linear spreading speeds associated to the asymptotic problems. Most
notably, when the profile of the coefficient diffusion is increasing we show
that there is an intermediate range for the forcing speed where spreading
actually occurs at a speed which is larger than the linear speed associated
with the homogeneous state around the position of the front. We complement our
study with the construction of strictly monotone traveling front solutions with
strong exponential decay near the unstable state when the profile of the
coefficient diffusion is decreasing and in the regime where the forcing speed
is precisely the selected spreading speed.
