Vetus Combi-Y 3000W - Connecting to shoreline issue..

I have the above until on a very recently purchased narrowboat. When attached to shoreline the batteries charge up to full.  All good there. I have 2 Lipos that feed 2 more leisure batteries. My questions are. Should the inverter stop running when the batteries are full? I do not wish to damage the Lithium batteries. Second question is. Can the inverter kick in when the batteries are getting low?