I find traders in different linked settlements share the same inventory,
but hold a separate stash of caps.
Don't know if that's how it should work.
The store trading is really rubbish, and needs to be fixed.
These stores are supposed to be trading with the settlers, with travelling caravans, and with the wider population. They should have good stock turnover, and plenty of variety. In practice, however, they just have a single shared stock which barely updates. If you buy most of the stock at one store, its gone from all the other settlements as well. Whats even more exasperating is if you pay by off-loading junk or other stuff you don't want. This is then offered back to you as "shop stock" for the next few days.
IMHO, if you've invested in 20 clothing emporiums or 20 gun shops, you should be able to count on going around and picking up 15 or so red dresses or whatever, or a few hundred shotgun carts - not just one red dress and ten carts because thats all the first shop had.
The "investment" option seems to make no difference at all. Whats the point of that?
I seem to have opposite of you? The stores all have the same cap count but will have some items that are different in their inventory. Been a godsend to get my civies all dressed