Your miners find some strange stones - they float to the ceiling of mines.
With those stones you can easily build an airship - just fill a sleek wooden hull with them and give the rowers some wing-paddles.
With a drag coefficient ~0.02, even under muscle power you can get some tens km/h, also you can use favorable wind.
Now you have one problem - there is no way to regulate lifting power. If you let the floating stone go it will go straight to space.
And the ship is constantly losing weight as rowers consume food and water. Also accidents happen - a paddle gets dropped, a rower goes overboard, some barrel gets broken and spills wine. Or you want to drop some flechettes on enemies' heads. And rowers can provide only so much thrust downwards...
You can just let some stones go - but they are very expensive. You can gather rainwater or dew from clouds - but that depends on weather.
Dropping ballast allows you to make the ship lighter. But you need antiballast - a way to make ship heavier in the middle of flight.
So what is an optimal way to increase the weight of, say, a 10 ton ship by at least 100 kg - I guess some reaction to bind oxygen or vapor that is accessible without industrial technology?
Compression of air is not a very optimal way to do it. Even with modern technology, a balloon weights several times more than air it contains. For preindustial one ratio would be order of magnitude worse.
Likewise, cool air - suppose we have a 10 ton 100 m3 airship with 300K on inside. If we cool it even to -70C then we would get about -30 kg, less than weight of adult. And making the ship bigger would increase drag and make it too slow under manual power.