Here's my take on an AssistShuffle team:
Skarmory @ Focus Sash
Keen Eye
Timid Nature (+Spe, -Atk)
252 HP | 4 Def | 252 Spe
- Stealth Rock
- Spikes
- Whirlwind
- Roar
Persian @ Choice Scarf
Limber
Timid Nature (+Spe, -Atk)
252 HP | 4 Def | 252 spe
- Assist
- Roar
- Sleep Talk
Purugly @ Choice Scarf
Thick Fat
Timid Nature (+Spe, -Atk)
252 HP | 4 Def | 252 Spe
- Assist
- Roar
- Sleep Talk
Drapion @ Lum Berry
Battle Armor
Bold Nature (+Def, -Atk)
252 HP | 252 Def | 4 SpDef
- Roar
- Whirlwind
- Toxic Spikes
- Thief
Blastoise @ Lum Berry
Torrent
Careful Nature (+SpDef, -SpAtk)
252 HP | 4 Def | 252 SpDef
- Counter
- Mirror Coat
- Roar
- Focus Punch
Swampert @ Lum Berry
Torrent
Impish Nature (+Def, -SpAtk)
252 HP | 252 Def | 4 SDef
- Counter
- Mirror Coat
- Roar
- Focus Punch
This team is rather straightforward. You open with Skarmory, who sets up SR and Spikes until knocked out or disabled. Once Skarm is done, you bring in Persian or Purugly (who sit at 541 and 532 speed, respectively) and start Assisting. There are ten moves that can be selected by either Assist user. Seven of these moves will phaze, while the other three will set up entry hazards. Toxic Spikes is obviously optional on Drapion, but I believe that the potential to set up poison damage is worth the 7% drop in shuffle success.
Anyways, because Assist can choose moves from fainted pokemon, you shouldn't worry about letting Skarm or Drapion bite the dust if it allows you to get another hazard in play or save an Assist user. Likewise, Blastoise or Swampert can be used to absorb priority hits (Scizor) and Counter back or manually phaze. Speaking of those two, they are your endgame lifeline.
Once you've worn down the enemy team or exhausted your supply of shufflers, the two water titans are your key to victory. Everyone's favorite turtle is equipped to take a special hit and fire back with Mirror Coat. Swampert takes care of a physical attack or two. Both also come with Focus Punch. This is mostly filler, but is potent if you predict a taunt or stat-upping move. Lum Berries can give you an extra turn to phaze or prevent horrible things like poison if you need to out-stall the last enemy pokemon.
I haven't tested this team at all, and it obviously might need some EV tweaking for specific situations. However, it looks usable in theory. Famous last words. :<