There are so many possible answers and the subject is so large!
<p>When the parametric family of distributions is some exponential family, there is this recent and very elegant paper:

 - [Consistency under Sampling of Exponential Random Graph Models][1]

Although it is focused on studying exponential random graph models, which are commonly used in modeling social networks (and other network models), the framework is quite general.


**Edit 1:** If the question is about which parametric families are closed under taking mixtures, then the following references are relevant

 - [Misiewicz et. al. (2005). Classes of measures closed under mixing and convolution. Weak stability][2]
 - [Teicher (1960). On the Mixture of Distributions][3]


  [1]: http://arxiv.org/abs/1111.3054
  [2]: http://webmail.impan.gov.pl/cgi-bin/sm/pdf?sm167-3-01
  [3]: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177705987