This is a re-post of the solution (or, more accurately, the work-around) that worked for me:
RCS messages sent from or shared with Android devices are not processed by Apple, and are not sent along via the Internet, your modem, and your router. (The senders of those messages often get an error message saying that you're "not on the network" or "not on line", as if your phone was not turned on.)
If my iPhone is on my desk, and is using Wi-Fi instead of the phone company's cellular signal, it won't receive RCS messages sent by the latter system. When I step outside, out of the router's range, or simply turn off the phone's Wi-Fi, I will get a cascade of all the unreceived messages that have been piling up.
This is probably just one of the ways RCS fails to work seamlessly with iPhones, but it might be the one that's giving you grief, in which case turning off Wi-Fi is the "fix". (Having to turn it on and off, depending on your needs, is what makes this a clunky workaround instead of a true solution.)