  In this article, we characterize isomorphism classes of Lefschetz fibrations
with multisections via their monodromy factorizations. We prove that two
Lefschetz fibrations with multisections are isomorphic if and only if their
monodromy factorizations in the relevant mapping class groups are related to
each other by a finite collection of modifications, which extend the well-known
Hurwitz equivalence. This in particular characterizes isomorphism classes of
Lefschetz pencils. We then show that, from simple relations in the mapping
class groups, one can derive new (and old) examples of Lefschetz fibrations
which cannot be written as fiber sums of blown-up pencils.
