Are the characters of each game traveling together throughout the entire story? Or do they arrive together, then split up and handle quests individually? And how do the DLC characters fit in? Like, if the BL2 characters are traveling as a group for the entire game, do we just assume that Gaige and Kreig have always been a part of that group, or do they come in part way through? Sorry if this is too much for one ask.
This is entirely up to the interpretation of the player.
However, what we do know is the group of Vault Hunters from Borderlands 2 are all together during the ending of Tiny Tina’s Assault on Dragon Keep. Also, six assassins are sent after the Vault Hunters during Sir Hammerlock vs. the Son of Crawmerax, and receive letters from their corresponding saviors upon the completion of “Victims of Vault Hunters”.
Finally, during The Holodome Onslaught, Axton and Gaige reveal they were previously hunting down Aurelia before Lilith called them back to Sanctuary after The Watcher saved Athena from execution.