Homogeneous coloured multipartite graphs

We classify the countable homogeneous coloured multipartite graphs with any finite number of parts. By Fraïssé’s Theorem this amounts to classifying the families F of pairwise non-embeddable finite coloured multipartite graphs for which the class F o r b ( F ) of multipartite graphs which forbid these is an amalgamation class. We show that once we understand such families F in the quadripartite case, things do not become any more complicated for larger numbers of parts.