I am 100% sure that in IPB3.x I as a moderator could move posts to another thread, but it seems impossible to do it in Invision Community. 

This seems very strange, what have I missed? 🤔

Found it. You have to give moderators access in the IPB control panel.

For those wondering, "Split" is the command which will allow you to split a post from a topic and either make a new topic out of it or add it to an existing topic.