Airline-filed PNRs, previous passport swipes linked to some databases, and passport stamps are the primary means by which border control personnel at airports manage to figure out a chunk of the travel history. They can attempt to do other things to try to figure out more travel history, but it may be far from a complete picture if you’re not carrying too much with you and/or you’re otherwise not in a position to willfully provide access to additional info so as to be an open door.
Even when the UK and Sweden were both in the EU, both countries’ officials were rather clueless about my complete cross-border travel history for travel to/from and within the EU. Even my own government with the most extensive set of resources and cross-border border control-related systems agreements would have a fractional window into my international travel history without that which I provide; and that is when the government has had levels of interest in and access to travel info well above what is accessible to border control agency employees working at any major airport to which I have been.
How much info does CBP share with other countries?
If I fly from A-B-USA and I’m in Global Entry, my US Passport has NO US entry stamp, would country A know I entered the USA?
Less than which other countries provide the US. But airline-filed APIS/PNR info can make its way to foreign governments even without CBP sharing. And with what’s in a PNR, it doesn’t take all that much to try to find out more.