Between 2013 and 2015, Marcus, Spielman and Srivastava wrote a sequence of papers where they famously solved the Kadison-Singer problem and proved the existence of Ramanujan graphs of all sizes. For the latter, they used a convolution of polynomials introduced by Walsh, which they showed to have surprising connections to free probability theory. This discovery gave rise to finite free probability, which studies polynomial convolutions from a free probability perspective. With the aim of unifying the results of Marcus, Spielman and Srivastava, and developing general machinery for deducing root bounds, we extend the framework of finite free probability to multivariate polynomials. We show that this extended framework has interesting parallels with the theory of freeness with amalgamation, and can potentially be used to obtain important results in diverse areas, ranging from algebraic combinatorics to operator theory. This is work in progress with Nikhil Srivastava.