Whenever the generator is running, the transfer switch relay will disconnect the shore power and the generator will supply all your power. When the generator stops, the relay will then reconnect shore power if you are plugged in. However it is not recommended that you do this, as repeated switching under load can cause arcing and eventual destruction of the transfer switch relay. Better to start and/or stop the generator with shore power unplugged or better yet, turned off at the breaker.
You can run the inverter at all times with no bad effects. However, if the coach is in storage and you don't need power for the fridge, you might want to turn off the inverters to protect the batteries from discharge.
Marc and Jill, Wellington FL
2013 Entegra Anthem 44SL
2018 Lincoln MKX