  We study the language-theoretic aspects of the word problem, in the sense of
Duncan & Gilman, of free products of semigroups and monoids. First, we provide
algebraic tools for studying classes of languages known as super-AFLs, which
generalise e.g. the context-free or the indexed languages. When $\mathcal{C}$
is a super-AFL closed under reversal, we prove that the semigroup (monoid) free
product of two semigroups (resp. monoids) with word problem in $\mathcal{C}$
also has word problem in $\mathcal{C}$. This recovers and generalises a recent
result by Brough, Cain & Pfeiffer that the class of context-free semigroups
(monoids) is closed under taking free products. As a group-theoretic corollary,
we deduce that the word problem of the (group) free product of two groups with
word problem in $\mathcal{C}$ is also in $\mathcal{C}$. As a particular case,
we find that the free product of two groups with indexed word problem has
indexed word problem.
