It seems, that if we uproot a Wheezewort from one Flower Vase (to plant it into another) it get destroyed/used/disappear.
Is this 'by design'?
How can I transport such a Wheezewort from one place to another?
I think you should submit a bug report on the bug tracker since it is not the intended behavior. I tried to do this on 3 different saves and had no problem relocating my chilly Wheezewort.