My Sims can ask to move in at another Sim's house, but it says "you need to let this person inspect the house you two will be living in, send your Sims home". Once they're home, the move-in option goes away... help?

Are you playing Freeplay? I can Currently only get to partner status in Freeplay - which means it's impossible to marry or move in with other sims otherwise.

But what I found out is you can try to "whistle" at any Sim you want to from any of the houses.

You do this by going to Simtracker first, then clicking on the whistle icon. Give it a go and see.

Also some other tips are that you have to have reached up to level 10 or 11 and the goals must be complete.

Then eventually you will see the get married option or the move in option will come by itself once you keep being romantic.