Not all airlines have rebooking agreements. Southwest is the same.
You are correct. American/United/Delta and maybe a few others like Alaska Air have ticket/baggage agreements. JetBlue Airways and American Airlines launched a partnership earlier this year allowing each carrier to sell one another’s flights but it's in limbo due to justice department litigation.
So, you can fly on let's say American from Columbus to Dallas and connect to a Delta flight the rest of the way to (for example) San Diego. But you can't do part of the trip on Southwest due to no baggage/ticket agreement.