For clarification regarding pottery planters:
Don't worry, planter racks still function the way they used to! They were not affected by this change.
It's timed. There's a bit of a random factor in how much it withers with each ageing tick, but if the planter is old enough to be picked, it should take multiple days in the cart for it to completely die off. Even if it takes an ageing tick or two while in the cart, as long as the plant is still alive, it will grow back to a normal, harvestable state once it's somewhere it can grow.
As of now, planters can grow:
On the ground
In planter racks
When placed on top of another object, such as a table.
If the planter is on your person (your inventory, a container in your inventory, a worn item like a backpack or satchel), it will neither wither nor age. Its age will be frozen. This is mostly to prevent them from withering while you're relocating them. This is the same as how they used to behave while in containers.
When inside literally anything else (a cart, a chest on the ground, a forge, a backpack on the ground), they will wither and die over the span of a few days.
Yes, dropping them into something like a chest or cart will cause them to wither away and die, letting you plant something else in them. However, it may take a few days for them to do so, so it may not be a convenient way to "unplant" them.
And yes, when placed on a table or a cupboard they will now act the same as they do in a rack or on the ground.