A 'router' could be doing anything - particularly NAT, which might involve any amount of bug-ridden messing with traffic...
One reason a device will send a RST is in response to receiving a packet for a closed socket.
It's hard to give a firm but general answer, because every possible perversion has been visited on TCP since its inception, and all sorts of people might be inserting RSTs in an attempt to block traffic. (Some 'national firewalls' work like this, for example.)