Most inverters do have low voltage protection, yes. Some at a fixed voltage, some adjustable.
Some also have high voltage limit as well, usually ~15 vdc. The high limit can sometimes be a issue with some battery multistage chargers that can touch to just over the limit during absorption mode.
Vince and Susan
2011 Tiffin Phaeton 40QTH (Cummins ISC/Freightliner)
Flat towing a modified 2005 Jeep (Rubicon Wrangler)
Previously a 2002 Fleetwood Pace Arrow 37A and a 1995 Safari Trek 2830.