Movement heavy fight tips?

So I just failed my first dungeon with my 3-box setup (Prot Warrior, Resto Shaman, Arcane Mage). I've been working my way up the Vanilla dungeons as I get the hang of multiboxing, and I made it to Blackfathom Deeps. The final boss, Aku'mai, just destroyed me.

I might just be overlooking something, but it seems to me to just require an excessive amount of movement - the tentacle adds start slamming the group if you don't move out of their path, and rocks begin falling on all my characters repeatedly and with very little warning. I tried broadcasting my strafe keys so I could quickly move everyone between spell casts, but it didn't work for long. Someone would eventually get stunned by a tentacle, leading to chaos. Doing this with 3 characters means the fight lasts a while too, but I'm hopeful I can do it with some patience.

Any suggestions for overcoming this fight, or movement-based fights in general? I'm aware that later dungeons, especially Legion and BFA ones, require more and more movement to pull off, so I'd like to figure out a good way to handle it. I'm using HotKeyNet, if that matters, but I can figure that part out. I'm looking more for general tips and strategy. Thanks!

So the dungeons are significantly harder than they used to be back in the day. If you're not running it with a full group it's a huge disadvantage and I'd say your doing great regardless.

Because a lot of fights are now movement heavy with mechanics mobile DPS is favored, which is primarily Beastmastery Hunters. You can still of course pull it off with other classes but your DPS begins to drop unless you weave in some instant casts here and there.

You just need movement keys for your non main group to weave in and out of these, so when you press strafe left only your Resto Sham/Arcane Mage go left while your prot war stays.
For me, I use WASD for main only movement, alt - WSAD for everyone but the main movement, and the arrow keys if I need to move the entire group.
Another nice trick is if the main doesn't have to move much, I'll swap to the healer and have the team follow that character and then dodging is a bit easier by just running around. If the tank needs to adjust, swap over and move and then swap back.
A good example of this type of fight is the last boss in Freehold. The tank doesn't have to do anything but tank, while the rest of the team is dodging artillery strikes, this keeps them clumped together well.

If you're using the pro setup you could try leading from the healer, that works way better for me in movement heavy fights. Its just so much easier to micromanage from a ranged.

You'll still need a movement method as Ellay suggests, which you'll find yourself using very often.

Ellay put it beautifully. For me my arrow keys are the rest of the group. I always back them out (or leave them behind at the beginning with a quick tap) and then strafe them around as needed. I always mouse move the main.

Thanks for the tips!
I managed to complete the dungeon, through a combination of adjustments:

I set up movement keys broadcast to just the ranged toons, as suggested.
I respecced the Mage to Fire and set up his bars to use Scorch instead of Fireball so I wasn't interrupting him constantly.
I created WeakAuras to show his Heating Up procs on all my windows so I could see and hit them while piloting from anyone.
I set up the Shaman's bars to use his instant spells; Riptide, Earth Shield and Healing Stream Totem, and the only one with a cast time he'd use was Healing Surge (no Chain Heal or Healing Wave).

I ended up mostly piloting from the Shaman, with the Mage stacked up on him the whole fight, strafing left and right. Took a few tries to get the movement timing right.