  We say that a matrix $P$ with non-negative entries majorizes another such
matrix $Q$ if there is a stochastic matrix $T$ such that $Q=TP$. We study
matrix majorization in large samples and in the catalytic regime in the case
where the columns of the matrices need not have equal support, as has been
assumed in earlier works. We focus on two cases: either there are no support
restrictions (except for requiring a non-empty intersection for the supports)
or the final column dominates the others. Using real-algebraic methods, we
identify sufficient and almost necessary conditions for majorization in large
samples or when using catalytic states under these support conditions. These
conditions are given in terms of multi-partite divergences that generalize the
R\'enyi divergences. We notice that varying support conditions dramatically
affect the relevant set of divergences. Our results find an application in the
theory of catalytic state transformation in quantum thermodynamics.
