It no doubt does depend somewhat on timeframe. In the mid-50s, it is pretty clear from consist information that there was some interchange. In ATSF consists there were shipments from NY and other eastern cities, and also some exchange with ACL. From ATSF consists, it almost appears as if cars were used like freight boxcars, loaded according to direction of travel and availability.
Looking at other roads, Wabash sent express from St. Louis to the west coast via UP connections at KC, to San Antonio via MKT at KC, and to Jersey City via Erie at Chicago, and cars were returning on the same routes.
The Dixie Route lines between Chicago and Florida (CEI, LN, NCStL, ACL, FEC) operated trains with a jumble of varied equipment. By the 60s, L&N cars might also occasionally show up on a Monon train. Likewise ACL-RFP-PRR-FEC intermixed head end cars on NY-FLA trains (as well as coaches, diners, etc).
Express reefers were a somewhat different animal- might be carrying highly perishable fruit in one direction, storage mail (or empty) in the other. REA had more of its own reefer cars, but might still need to use railroad owned cars occasionally, and I think these did operate in more of a pool arrangement- GN or ATSF or PRR would be happy to see unused cars earning some revenue, and while virtually every road had its own baggage cars, only a relative few had refrigerator cars with steam lines and high speed trucks for use on passenger trains. Even the CN 8-hatch cars roamed a bit on occasion.
One other interesting area is RPO cars. While these seem to be "captive" on most runs, there were occasional exceptions (CN on the CV, as discussed elsewhere on threads about the new MT car) for leased cars or subsidiaries. Also, some mail routes were by definition joint ventures between 2 or more railroads (for instance, at one point in time in the 1930s, CEI L&N and NC&StL had a mail contract from Chicago to Atlanta)- so the RPO cars must have been pooled in some fashion on those routes.
I would note that it appears that when possible, foreign road baggage or express cars were moved in secondary and night trains. Unfortunately, it means few pictures- people took lots of photos of the Super Chief, precious few of the Fast Mail.