  On relational structures and on polymodal logics, we describe operations
which preserve local tabularity. This provides new sufficient semantic and
axiomatic conditions for local tabularity of a modal logic. The main results
are the following.
  We show that local tabularity does not depend on reflexivity. Namely, given a
class $\mathcal{F}$ of frames, consider the class $\mathcal{F}^\mathrm{r}$ of
frames, where the reflexive closure operation was applied to each relation in
every frame in $\mathcal{F}$. We show that if the logic of
$\mathcal{F}^\mathrm{r}$ is locally tabular, then the logic of $\mathcal{F}$ is
locally tabular as well.
  Then we consider the operation of sum on Kripke frames, where a family of
frames-summands is indexed by elements of another frame. We show that if both
the logic of indices and the logic of summands are locally tabular, then the
logic of corresponding sums is also locally tabular.
  Finally, using the previous theorem, we describe an operation on logics that
preserves local tabularity: we provide a set of formulas such that the
extension of the fusion of two canonical locally tabular logics with these
formulas is locally tabular.
