Here is a link to a paper examining the inner workings, leadership, response capability, reporting mechanisms, mission profiles etc. of the Russian emergencies ministry under (now) Vice President Sergei Shoigu:
The article mentions prospects for working with NATO in disaster response. Here is a NATO press release examining a Memorandum of Understanding formed after the US Foreign Military Studies Office produced the former report:
Scholars have examined the view that environmental catastrophes present a "security risk" for nations suffering through them. Criminal statistics from the aftermath of, for example, the Venezuelan floods, sustains the conclusion that indeed, they are. It is no surprise, then, that increasingly, military forces are called out to deal with severe 'natural' disasters. Military response to catastrophic events enjoys wide public support.