  In a previous paper, we studied the Christoffel transforms of multiple
orthogonal polynomials by means of adding a finitely supported measure to the
multiple orthogonality system. This approach was able to handle the Christoffel
transforms of the form $(\Phi\mu_1,\dots,\Phi\mu_r)$ for a polynomial $\Phi$,
where $\Phi\mu_j$ is the linear functional defined by $$f(x)\mapsto \int
f(x)\Phi(x)d\mu_j(x).$$ For these systems we derived determinantal formulas
generalizing Christoffel's classical theorem. In the current paper, we
generalize these formulas to consider the case of rational perturbations
$$\Big(\frac{\Phi_1}{\Psi_{1}} \mu_1,\dots,\frac{\Phi_r}{\Psi_r}\mu_r\Big),$$
for any polynomials $\Phi_1,\dots,\Phi_r$ and $\Psi_1,\dots,\Psi_r$. This
includes the general Christoffel transforms $(\Phi_1\mu_1,\dots,\Phi_r\mu_r)$
with $r$ arbitrary polynomials {$\Phi_1,\dots,\Phi_r$,} as well as the
analogous Geronimus transforms. This generalizes a theorem of Uvarov to the
multiple orthogonality setting. We allow zeros of the numerators and
denominators to overlap which permits addition of pure point measure. The
formulas are derived for multiple orthogonal polynomials of type I and type II
for any multi-index.
