Update: At IAD, the Egyptair check-in agent declined to tag the bags all the way to BOM. Her supervisor also declined stating that they don't have an agreement with VIstara. I mentioned what B747-437B said about both airlines using the same Amadeus system. They still declined. I was like okay, I will pick up bags in DXB and re-checkin. Then as she was trying to print the tags, my best guess is that the system was going to print tags all the way to BOM. So she called her supervisor so that he could show her how to print only until DXB. I raised my concern again. That's when the supervisor asked me to talk to their Station Manager. The station manager asked if Vistara is low cost airline. I said no. Then he was on the phone with someone. Then he said ok we will try to print the tags. If they print until BOM, then it is okay if not then he can't do anything. Furthermore, if Vistara were to ask for extra baggage fees then it is my responsibility. I was ok, lets try printing. Sure enough, the tags came out to BOM. At DXB I had to go to transfer desk where they took all my tag numbers and transfered my bags to vistara and issued final boarding passes just as B747-437B and plunet said.

Thanks guys for your suggestions. Without which, it would have been a mess trying to deal with bag transfers and 2 sleepy kids