If you want to optimize your gameplay, switching improvements is crucial. For example when running SP you can switch alot of farms and mines to workshops, windmills and watermills - same overall output and some commerce on top.
On Grassland:
Farm: +4 food
Mine (+railroad): +4H/1F
Watermill: 3F/2H/2C
Windmill: 2F/2H/2C
Ofc that's only true for when you can convert a riverside farm and a hill at the same time, otherwise you might mess up your stable configuration for that city. Also only works when you have rivers, and there's only a real advantage when you have state property and electricity - but then, this configuration actually is superior, even more when you're financial.
Before that, watermills are simply the best improvement for riverside plains, even before you have rep. parts. Windmills suck before rep. parts, though, but might be an option if you're financial and are desperate for food.
And Workshops ... well, think about it: in a 18 grassland tile city without ANY hills and ANY food you could theoreticaly have a stable configuration with 54H / turn. Yes, it'd need forever to grow into those tiles without farms, but in the lategame you can have workerstacks of 4 workers and switch farms into workshops in 1 turn -> grow into the optimal size and switch to mass workshops afterwards. It's a common strategy especially for warmongering empires that didn't have the time to grow cottages, and ofc for every space race VC where you'll need ~8 strong hammer cities.
Lumbermills are somewhat a special case, normally all forests should be gone long before you can even build them, at least in your part of the world. If you conquer AI cities you might find some forests unchopped, but it's questionable if you really want to keep the forests for lumbermills - they need railroad to be competitive with workshops. Better chop them for infrastructure in your newly conquered cities if you ask me. The only exception here might be:
a) you're really, really desperate for health and can't afford the time to build or research any health buildings. But you have to be really, i mean REALLY desperate. Haven't seen something like that in any game.
b) tundra forests on non-river tiles. Sometimes your city hasn't got any hammer tiles otherwise, so it might be clever to leave those forests.
c) you want to build National Park in that city and bring those forests to some use in the meantime, so build lumbermills first and switch to preserves later. But honestly, i never build that building, it seems kinda useless to me. Maybe if i get a lone island very late in the game or something, but even then it's just for fun and won't contribute for winning the game too much, rather delay it most likely.
tl;dr: workshops, windmills, watermills -> yes, lumbermills -> no.