This initiative is called "Moms Demand Action". It is politically independent and not affiliated with any political party. Interesting and surprising enough the aims of that initiative are very moderate from an European point of view. So why should only assault weapons be banned? Nevertheless its make sense for sure to support these grass root movements!
From their webpage:
2) Require background checks for all gun and ammunition purchases.
3) Report the sale of large quantities of ammunition to the ATF, and ban online sales of ammunition.
4) Make gun trafficking a federal crime with serious criminal penalties.
5) Counter gun industry lobbyists’ efforts to weaken gun laws at the state level.