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 FF of pairwise non-embeddable finite coloured multipartite graphs for which the class Forb(F)Forb(F) of multipartite graphs which forbid these is an amalgamation class. We show that once we understand such families FF in the quadripartite case, things do not become any more complicated for larger numbers of parts.