  In this paper, we study local regularity properties of minimizers of nonlocal
variational functionals with variable exponents and weak solutions to the
corresponding Euler--Lagrange equations. We show that weak solutions are
locally bounded when the variable exponent $p$ is only assumed to be continuous
and bounded. Furthermore, we prove that bounded weak solutions are locally
H\"older continuous under some additional assumptions on $p$. On the one hand,
the class of admissible exponents is assumed to satisfy a log-H\"older-type
condition inside the domain, which is essential even in the case of local
equations. On the other hand, since we are concerned with nonlocal problems, we
need an additional assumption on $p$ outside the domain.
