In Hirschhorn's Model categories and their localizations he gives a sufficient condition to induce a cofibrantly generated model structure on a category $\mathcal{N}$, given an adjoint pair of morphisms $\mathcal{F:M\rightleftarrows N:U}$ (where $\mathcal{M}$ is a cofibrantly generated model category) such that the adjoint functors induce a Quillen equivalence. Using Kan's characterisation of cofibrantly generated model categories.
Is there a way to make a similar statement about combinatorial model categories along the lines of Jeff Smith's theorem?
My main motivation for this is to see whether I can use the diagonal functor from the category of $n$-simplicial sets to the category of simplicial sets to define a combinatorial model structure on $n$-simplicial sets.
Yes, there is such a criterion for combinatorial model categories, though it is quite similar to Hirschhorn's: the only difference is that the smallness conditions (11.3.2.(1) in Hirschhorn) can now be dropped.
The transferred model structure on bisimplicial sets (and multisimplicial sets are no different) is the subject of Moerdijk's paper “Bisimplicial sets and the group-completion theorem”.