  We characterize the semigroups of composition operators that are strongly
continuous on the mixed norm spaces $H(p,q,\alpha)$. First, we study the
separable spaces $H(p,q,\alpha)$ with $q<\infty,$ that behave as the Hardy and
Bergman spaces. In the second part we deal with the spaces
$H(p,\infty,\alpha),$ where polynomials are not dense. To undertake this study,
we introduce the integral operators, characterize its boundedness and
compactness, and use its properties to find the maximal closed linear subspace
of $H(p,\infty,\alpha)$ in which the semigroups are strongly continuous. In
particular, we obtain that this maximal space is either $H(p,0,\alpha)$ or
non-separable, being this result the deepest one in the paper.
