Melon Farmer / Lemon Drinker
Perospero is clearly angry at Beast Pirates as much he is at Strawhats.
He is also clearly not going to honour the alliance.
So I think Perospero will reach to top and find out Luffy and Yamato forming alliance. He may see them talking together or fighting together. Given he already doesn't like the alliance, he will immediately conclude that Kaido is backstabbing Big Mom Pirates by secretly sending his heir to make alliance with Strawhat Pirates. He will naturally act on it and try to assassinate Luffy and Yamato. He might land successful hit on Yamato given surise factor. Luffy will attack Perospero in return for hurting his nakama. King also might attack Perospero.
Incident news will travel through word of mouth and getting corrupted over the course. Both Kaido and Big Mom will hear it from biased sources making both of them think that other one is trying to kill them. Paranoia will erase any trust made. Any action even harmless one would look as threat to them. Sooner or later Kaido and Big Mom will come to blow. They will fight and weaken each other. That is how both Yonkou who are clearly beyond alliance's power to fight will be damaged enough to lose against it.
Basically cause of fall of two powerful Yonkou would be bitterness of one Perospero. He will be the catalyst that sets their downfall in motion.
What do you guys think?
All credit goes to Amol from nf