It's fairly clear that this is the polite thing to do whereas attempting to invite myself to theirs would be rude

To be fair, I don't tend to invite myself to other people's houses for dinner, but on the whole (if carefully picked) they don't mind demands for tea...

I suspect my inclination would be to open with "we should do dinner some time" and see where the conversation went from there, and I suspect I'd also end up more-or-less quits (in terms of travelling etc) with anyone I liked that much (I are fail at socialising), but I have very little experience of this manner of thing...