A Pokemon with moves like Spore and Sleep Powder work great because a Pokemon that is asleep has double the capture rate. Thunder Wave also works well because it induces Paralysis, where the Pokemon has 1.5x the rate of capture. Both of these status conditions can make Pokemon not attack, so it is great for capture.
Another great move, as you said, is False Swipe, as it always leaves the Pokemon with 1 HP, making it have the highest capture rate it possibly can.
3 Pokemon can learn these moves: Paras, Parasect, and Shroomish. Here are sample movesets for capture:
Parasect (Defensive Capturing Pokemon) @ Leftovers
Relaxed Nature (^Def vSpd)
Ability: Effect Spore
EV Spread: 128 SpDef, 252 HP, 128 Def
Protect (For stalling)
Stun Spore (For Pokemon with Insomnia or other sleep-canceling abilities)
Breloom (Offensive Capturing Pokemon) @ Leftovers
Jolly Nature (^Spd vSpAtk)
Ability: Technician/Effect Spore
EV Spread: 252 Atk, 252 Speed, 4 HP
Force Palm (For chance of Paralysis for Pokemon with sleep-canceling abilities)
Mach Punch (If ability is Technician)