For a map $f: Y \rightarrow X$ smooth outside a simple normal crossing divisor $B=\sum_iB_i$, do we know of similar local freeness property for higher direct image of relative canonical sheaf plus a fractional divisor supported on the preimage of the branched locus? i.e. is $R^qf_*M$ also locally free, where M is cartier and is numerically equivalent to $K_{Y/X} + \sum_id_iB'_i$? Here, $B' = \sum_iB'_i= (f^*B)_{red}$ and $\lfloor \sum_id_iB'_i \rfloor = 0$
A relatively unrelated question: do we know anything about how the preimage of the branched locus could look like?