Doing things even when you aren't sure if they are your responsibility is precisely what makes you a responsible person. Encouraging everyone around you to do trivial things like bringing the shopping cart back is actually a big deal. And the most effective way to do this is to just do it. Park as far as you possibly can from the store and walk that thing for a solid 3 minutes both ways. Find one with a really noisy wheel. Make a whole production out of it. Leading by example is unbelievably effective if you lean into it just a little bit.
But this could be anything. If you see a customer waiting on a response and you know exactly what they need to be told, go tell them. Don't make them wait until next Thursday when their CSR gets back from vacation. If you aren't sure about the email, draft it, send it to your boss/peer/etc. for review. About 50% of the time they'll say "yep looks good please send" the other half they'll just take care of it right then and CC you on it. Either way you come out looking good.
The "not my job" crowd is simply not thinking ahead at all. You don't get paid more money the microsecond you go above and beyond. It takes persistent investment in this bucket before someone with actual power decides to help you out. There's not a progress bar or quest log you can review. It just happens. One day you get a phone call and that's that. There's no lead up or anticipation most of the time. You just have to be good all the time and expect that you are being observed. That's the only thing I've found that works over strategic timescales.