what is your explanation for a refugee that keeps walking through safe country after safe country before finally finding the specific country they accept?

Huh? Because it is safe obviously.
Everyone wants a better chance of better living.
And refugees are already moving.
Since they are already moving why not move to a country with better chance of better living?
If they hear “so-and-so country has better chance of better living” of course they are going to choose that, again, because they are already moving.