  We refine prior bounds on how the multivariable signature and the nullity of
a link change under link cobordisms. The formula generalizes a series of
results about the 4-genus having their origins in the Murasugi-Tristram
inequality, and at the same time extends previously known results about
concordance invariance of the signature to a bigger set of allowed variables.
Finally, we show that the multivariable signature and nullity are also
invariant under $0.5$-solvable cobordism.
