We study the asymptotic pole distribution and the convergence in capacity of AAK-type meromorphic approximants as well as 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 $\mu$ a complex measure with compact regular support included in $\R$, whose argument has bounded variation on the support.