  We classify the countable homogeneous coloured multipartite graphs with any
finite number of parts. By Fraisse's Theorem this amounts to classifying the
families F of pairwise non-embeddable finite coloured multipartite graphs for
which the class Forb(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.
