  In this paper, we give a simple description of the deformations of a map
between two smooth curves with partially prescribed branching, in the cases
that both curves are fixed, and that the source is allowed to vary. Both
descriptions work equally well in the tame or wild case. We then apply this
result to obtain a positive-characteristic Brill-Noether-type result for
ramified maps from general curves to the projective line, which even holds for
wild ramification indices. Lastly, in the special case of rational functions on
the projective line, we examine what we can say as a result about families of
wildly ramified maps.
