Hello,
AMD users go into Radeon software game settings and enable: enhanced sync
Also enable Vsync ingame settings as well. These should fix the mid/late game lag for sure.
Besides keep in mind that Windows 10 Home has severe limitations and contributes to games not running at the best performance so it’s a must to upgrade to windows 10 Pro. I have had two users pm me about aoe2 on voobly having a 2 - 3 delay to start and another claiming that aoe2 de lags for mid/late game, guess what? both of them were running on windows 10 home as soon as they the first user upgraded to Pro the delay disappeared while the second user It partially fixed some issues for him until he checked the enhanced sync in AMD software.
As a moderator on r/aoe2, I have read lots and lots threads where issues mostly arise on AMD graphics cards. Hopefully the tips above covers it.
I have a section on my blog post where you can find related tips to this topic: Tips To Fix FPS Drops and Crashes Mid Game