You have a choice of rules to use (or not):
Carrying capacity, it's 15 lb/STR, if the fighter's gear & the other character & his gear exceed that, he's pushding/dragging and his move drops to 5' - otherwise fine, this is the simple default for carrying stuff.
Encumbrance ("Variant"): Up to 5lb/STR he's fine, but it's unlikely a medium ally is under 100lb, which'd be the limit for 20 STR. Up to twice that, he loses 10' of movement, over twice that he loses 20' of movement and gets disadvantage on a bunch of stuff.
Moving a Grapple - you can probably dispense with the checks for a willing ally, but moving a grapple is at 1/2 speed.
All the above presumably require at least a free hand. You could require an action or just an attack, for the grapple (or an action for any of 'em, really, it's totally up to you).
Another thing you might go with, not a rule at all, but you're free to do what you like: Allow the ally he 'grabs' to spend his Reaction to go with him - instead of picking him up, you're just pulling him along, willingly. On his turn, that ally has already used it's move.
Whatever you do, decide quick, and narrate success/failure with confidence and flair!