Think this might be referring to the forced marching traits. If you repeatedly use up all the movement points of a family member they will get traits that will slow them down, do this too much and they'll be barely able move. Also there are movement restrictions in certain season depending on where you are (summer in desert areas, winter in temperate).
These only apply to family members, captains and suchlike are not affected as they cannot recieve traits.
edit: resting them for a turn or too allows them to recover
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