Up until 1975, all of Marvel’s western heroes existed in a separate reality from the modern Marvel superheroes. That is until part of the Avengers went back in time to encounter Kid Colt, Night Rider, the Rawhide Kid, and the Two Gun Kid in issue #141-#142 of their namesake title. Later in issue #147, Two Gun returns with Hawkeye to the future for a few adventures, some of which carry over in Champions #11, Ghost Rider #27, Marvel Tales #100, and Daredevil #215 (as a parallel flashback story). There is a second major time travel story when Ironman leads the Avengers back once again to the wild west in issue #18 of the West Coast Avengers (1987). Recently, Red Wolf was spirited away into the present in a six issue series in 2016 and apparently, he is remaining there for the time being.
DC also occasionally tosses modern super heroes like Batman back in time and most recently, Booster Gold who met Jonah Hex who was then subsequently thrown into the present day for a year of his New 52 series. Back in the 1970’s, Jonah Hex appeared in the Justice League comic a few time in issues # . Hex, was once again thrown into the far future as told in the short lived Hex series 1985-1986. In several of the DC cartoon TV series, like Justice League, the modern greats have met the wild west gun hands as well.