As I said earlier, they don't really ever "break down". The problem I experienced is that combat can get grindy in Tier 3 and Tier 4. Mostly because monsters lack sufficient offensive firepower. (There are exceptions, of course.) Monsters don't get enough attacks and their hits are underpowered. I'd suggest dialing up damage by a factor of 2x or 3x, easily. I would also dial hit points up or down -- up for boss monsters, down for the rest.
To give you an example of what I mean, a pit fiend is a pretty damn good creature. It doesn't need much modification. If PCs faced it in Tier 3, it would be a boss monster. I'd run it as written (perhaps adding some legendary resistances/actions). In Tier 4 you might need a squad of pit fiends to challenge the PCs. In that case, I'd keep their offensive numbers as-is, but reduce their hit points by...one third?...so the encounter resolved more quickly.
Unfortunately, many other high CR monsters are not as formidable, so they would need tuning on the offensive side.