My Catalyst readers kept asking about Alex, the main character exercising more, especially as his health started failing, but he was traveling so much I just couldn't arrange it easily.
I haven't actually read the Catalyst series yet -- it's on my "to do list", but so are about two dozen other books, and growing -- but a lot depends on how the travel is taking place, and what the goal of it is, bother in the short-term and the long-term.
If he's travelling by plane or coach (stage- or bus), he could do simple exercises like push-ups, jogging in place, and stretching in the aisles. If driving, he'd need the occasional rest break anyway, so he could squeeze in a bit of exercise then.
In a lower tech. setting, including post-apocalyptic, there's actually a fair bit of free time. In long term travel situations, horses need a lot of rest in order to stay healthy; it's generally advised to only ride about 6 hours per day, preferably in the morning before it starts to get hot, so a character has several hours in the afternoon and evening to hunt, scout, exercise, read, and so on.
Now obviously none of this means that the character necessarily should be exercising, it may simply not be in his character to do so. I'm just pointing out that no matter how busy a person is, there is always time to exercise if they choose to make it a priority.
This applies to real life, not just writing, and yes, I have my own issues with not prioritising exercise as often as I should, but there are always ways to squeeze some in. Back when I was working from home, I'd do a bit of stretching every time I stood up to refresh my coffee cup.