In this work, we describe a method to construct the generic braid monodromy of the
preimage of a curve by a Kummer cover. This method is interesting since it combines
two techniques, namely, the construction of a highly non-generic braid monodromy and
a systematic method to go from a non-generic to a generic braid monodromy. The latter
process, called generification, is independent from Kummer covers,
and it can be applied in more general circumstances since non-generic braid
monodromies appear more naturally and are oftentimes much easier to compute. Explicit
examples are computed using these techniques.