No, this is not directly addressed on the TV series.
However, two of the main characters, travelers 3468 and 3569 whom come occupy the bodies of Special FBI Agent Grant MacLaren and Marcy Warton (respectively) do come to remember some of their host's memories.
Traveler 3468 nearly dies after a plan crash and in which he is saved by futuristic nanite-repair technology that somehow allows him access some of Grant's memories, while in surgery.
Marcy goes to some trouble to try to remember what she'd forgotten
when she was overwritten. She connects herself to a monitor and bathes
in a bath of ice attempting to bring herself to the edge of death
since almost dying caused Grant to witness his host's memories.
As for traveler 3569, she gets a bit of a double-whammy, due to her host's medical condition. She's tricked into being reset with her more recent memories, the memories required since she inhabited Marcy's body, erased. After doing so, she finds she can't live with not knowing what the previous version of herself was like and takes extreme life-threatening measures to try and recover memories like 3468 does. However, she ends up retrieving many of Marcy's memories instead, which result in a plot twist revolving around the first traveler, 001 also known as Vincent Ingram.
In addition, even though we never get his previous name, traveler 0015 (Trevor Holden), is one of the oldest known travelers and has mentioned his family, including two sons who died of old age in the future and in one episode, we get glimpse of a previous body he was in along with his wife. This could be significant, because 0015 chooses not to get extremely intimate with anyone from the 21st century, even pushing fellow traveler Grace aside, but yet he was the one who was generally the most compassionate towards everyone and could be seen as the one to teach the others that. The other travelers on their team didn't seem to have families with children in the future, like 0015 did.
Also from Executive Producer/Creator Brad Wright in an interview prior to season 3 about the importance of 'the here and now'.
“I honestly never intended to go to the future because the characters
are trapped in the here and now and so should the audience be, in a
sense. Thematically, as much as it’s about people from the future,
it’s really about the here and now, and we’ve got to get our shit
together, because look what could happen.”
One point I think the show was making was that, despite that these people started out as essentially different people, that many of the travelers on this specific team came to love the people in their host's lives. And although I think there could be more to it, given the way memory/consciousness transference was beginning to be played with, I think the point was that these travelers were more compassionate than most and it's what thematically rendered their team more successful and set them apart from other teams.