We study diagonal multipoint Padé approximants to functions of the form \[F(z) = \int\frac{d\mes(t)}{z-t}+R(z),\] where $R$ is a rational function and $\mes$ is a complex measure with compact regular support included in $\R$, whose argument has bounded variation on the support. Assuming that interpolation sets are such that their normalized counting measures converge sufficiently fast in the weak-star sense to some conjugate-symmetric distribution $\sigma$, we show that the counting measures of poles of the approximants converge to $\widehat\sigma$, the balayage of $\sigma$ onto the support of $\mes$, in the weak$^*$ sense, that the approximants themselves converge in capacity to $F$ outside the support of $\mes$, and that the poles of $R$ attract at least as many poles of the approximants as their multiplicity and not much more.