We use techniques from finite free probability to analyze matrix processes related to eigenvalues, singular values, and generalized singular values of random matrices. The models we use are quite basic and the analysis consists entirely of expected characteristic polynomials. A number of our results match known results in random matrix theory, however our main result (regarding generalized singular values) seems to be more general than any of the standard random matrix processes (Hermite/Laguerre/Jacobi) in the field. To test this, we perform a series of simulations of this new process that, on the one hand, confirms that this process can exhibit behavior not seen in the standard random matrix processes, but on the other hand provides evidence that the true behavior is captured quite well by our techniques. This, coupled with the fact that we are able to compute the same statistics for this new model that we are for the standard models, suggests that further investigation could be both interesting and fruitful.