When can a sum and integral be interchanged for distributions?

Question

There is quite a lot of information about exchanging a limit and integral for functions and the condition for that is known to be the dominated convergence. Is there anything similar which can be established for distributions? Meaning, is it possible to establish a rule for when $$\int_{a}^{b}\lim_{\varepsilon\rightarrow0}f(x,\,\varepsilon)\,dD(x,\,\varepsilon)=\lim_{\varepsilon\rightarrow0}\int_{a}^{b}f(x,\,\varepsilon)\,dD(x,\,\varepsilon)$$ if $D(x,\,\varepsilon)$ denotes a distribution and $f(x,\,\varepsilon)$ is a test function? I was able to find examples in which this rule works and in which it fails, but could not find a place which discuss that in the literature.