Apparently, according to the in-game manual, the title is the case - "lottery winners" can only bring stuff from their own carriages.

However, this appears to not be the case - my "lottery winners" have access to all scavenging equipment, no matter where it is on the train.

I hope this is intentional and not a bug, because having to move yet more stuff around just to ensure the lottery winners have access to the right equipment would be a massive pain in the ass.

The only issue I found - apart from the manual being incorrect - is that occasionally you'll get some event where one carriage wants something from another involving these tools. I had one carriage request "scavenging equipment" from another carriage so their "lottery winners" would have better chances! Which is of course pointless since it makes no difference where this equipment is at the moment!

I'm so hoping this isn't a bug and it won't get "fixed" some time in the future - apart from the event being removed of course...