  We consider cocycles obtained by composing sequences of transfer operators
with positive weights, associated with uniformly expanding maps (possibly
having countably many branches) and depending upon parameters. Assuming $C^k$
regularity with respect to coordinates and parameters, we show that when the
sequence is picked within a certain uniform family the top characteristic
exponent and generator of top Oseledets space of the cocycle are $C^{k-1}$ in
parameters. As applications, we obtain a linear response formula for the
equivariant measure associated with random products of uniformly expanding
maps, and we study the regularity of the Hausdorff dimension of a repeller
associated with random compositions of one-dimensional cookie-cutters.
