This answer should perhaps be posted as a comment, since the whole topic was already debated in this MathOverflow Q&A. However i briefly restate the main result here and leave the above link for further detals: a definitive answer is Cafiero's convergence theorem (see [1]) which, roughly states that$\DeclareMathOperator{\Dm}{\operatorname{d\!}}$ $$ \int_X f_n\Dm\mu_n - \int_X f\Dm\mu_n \to 0 \iff \text{$(f_n\cdot\mu_n)_{n\geq 1}$ is uniformly exaustive.} $$ Note that this necessary and sufficient condition is not very well known, even in the circles of measure theorists.
Reference
[1] Cafiero, F. (1953), "Sul passaggio al limite sotto il segno d'integrale per successioni d'integrali di Stieltjes-Lebesgue negli spazi astratti, con masse variabili con gli integrandi" [On the passage to the limit under the sign of integral for sequences of Stieltjes–Lebesgue integrals in abstract spaces, with masses varying jointly with integrands], Rendiconti del Seminario Matematico della Università di Padova (in Italian), 22: 223–245, MR 0057951, Zbl 0052.05003.