We define a natural partial exchangeability assumption for multi-class systems with Polish state spaces, under which we obtain results extending those for exchangeable systems: the conditional law of a finite system given the vector of the empirical measures of its classes corresponds to independent uniform permutations within classes, and the convergence in law of this vector is equivalent to that of the system. A corollary is that convergence within each class to an infinite i.i.d. system implies asymptotic independence between different classes. We also extend the Hewitt-Savage 0-1 Law.