It might be a good idea to list the weaknesses of the Assistshuffle strategy together with wasy to solve them:
-Hail/Sandstorm: because the Assistshuffling pokémon carries choice scarf, it'll only have a maximum of 16 turns to attack before it dies to the residual damage. On top of that, Tyranitar is quite common.
Solution: This is probably the biggest threat to Assistshuffling teams. Sadly, weather moves are selectable through Assist, and even if you would be able to clear the skies, the opponent can simply switch in Tyranitar again. The best thing you can do is pack 3 Assistshufflers and hope that 48 turns is enough.
-Priority Moves: Even when using max HP, Persian, Purugly and Smeargle stay very frial and get 2HKOd by for example CB Scizor's Bullet Punch.
Solution: Counter isn't selectable through Assist and a good way to get rid of priority users, as long as Counter actually OHKOs. Your opponent won't Bullet Punch your Swampert again after he sees it could be his only chance of survival against the strategy. Reflect, although selectable through Assist, turns a lot of Priority moves from 2HKOs to 3 or even 4HKOs.
-The final pokémon: It's impossible to shuffle the final pokémon to death because it can't be phazed.
Solution: Unselectables like Counter, Mirror Coat and especially Destiny Bond can quickly finish off the final opponent. Also, seeing as it probably already has taken a lot of damage, unselectables like Chatter and Focus Punch can probably finish them off without too much Attack investment. Perish song, although selectable, can net the win 100% of the time as long as you're able to stall out for 3 turns.
-Unphazable pokémon: Suction Cups and to a lesser extend Soundproof Pokémon cannot be phazed and therefore are immune to the strategy.
Solution: Again, unselectables can do the job here, but are harder to pull off because these pokémon will likely be at near-full HP. Luckily, these pokémon are quite rare. Soundproof pokémon can still be phazed with Whirlwind, so you'll just need some luck to defeat them.
-Lack of entry hazard: If the opponent prevents you from setting up entry hazard or Rapid Spins it away, you can phaze all you want, but won't get anything out of it.
Solution: Rapid Spinners are quite rare nowadays, and won't get the time to Rapid Spin anyways if you play well enough. However, setting up the first time is the biggest problem. A lot of leads are designed to prevent setting up SR, and a lot of moves that would ease setting up entry hazard like Taunt can't be choosen due to Assist being able to select them. You could make Skarmory scarfed and simply outrun these leads, but you'll have to sacrifice something for the second setup. Luckily, even if your Skarmory faints, Assist can still select Spikes and SR and if it doesn't it will still prevent the opponent from attacking. As long as your opponent doesn't carry priority moves or Sandstorm/Hail, you can simply stall out until you setup some entry hazard. You will probably lose an Assistshuffler this way, though. If you're really afraid of Rapid Spin, you could pack a Ghost to block it, but there aren't many Ghosts that can use moves that could be useful to the team otherwise.
Any more we could add?