Dispersive processes with a time dependent diffusivity appear in a plethora of physical systems. Most often a solution is attained for a predefined form of diffusion coefficient D(t). Here existence of pulse solutions with an arbitrary time dependence thereof is proved for the Gierer-Meinhardt model with three types of transport: regular diffusion, sub-diffusion and L´evy flights. Admission of a solution of the classical pulse shape, but for an unencumbered form of D(t) is a valuable property that allows to study phenomena of the ilk observed in various ostensibly unrelated applications. Closed form solutions are obtained for some pulse constellations. Transitions between periods of nearly constant diffusivities trigger respective cross-over between counterpart solutions known for a constant diffusivity, thereupon exhibiting otherwise unattainable behaviour, qualitatively reconstructing observable evolution peculiarities of tagged molecular structures, such as essential slowing down or speeding up during various stages of motion, inexplicable with a single constant diffusion coefficient.
