After a player picks up the enemy flag and begins to receive damage, he will have to take into consideration his Grip. If his Grip is reduced to zero (by receiving damage or going into Overdrive), the player will fumble and automatically drop the enemy flag. The player can pick up the flag again but at a reduced Grip value, so a second fumble will be likely if he takes minimal damage or tries to OD.
Grip can regenerate on its own, albeit very slowly. However, if the flag is passed to a teammate, the Grip regeneration is vastly increased, and if the enemy player retrieves the flag, the Grip is reset to full strength.
*Flagstand bonus: Applies to any player that has the flag and is near a flagstand. Think of it as a standard bearer bonus, applies to if the player is at the enemy flag stand or his own. Flagstand bonus at his own base would be a boon to HoFs as they won't need to be concerned about their Grip strength as much when they're in spawn. Flagstand bonus is also beneficial to a capper as it allows them to take damage/overdrive after picking up the flag without worrying about their Grip strength being next to nothing as they try to make a getaway home. (Spamming the flagstand to drain Grip against cappers won't be a viable tactic with the FS bonus).
Passive items: Titan's Grip, Increased Grip strength.
Effect on Rabbiting: Likely hamper rabbits as the rabbit tries to solo the flag around the map, receiving damage and losing Grip. Possibly force rabbits into passing to teammates to regain Grip strength/respawn, most likely will just have rabbits die sooner.
Effect on heavy capping: Likely will nerf heavy capping, as they're a big target and rely on insta-OD (which would drain Grip stamina), Heavy cappers will need to rely on effective passing to get the flag away from the enemy.