One trick with drywall mud is to leave a complete layer of water on top for long term storage. As long as you carefully poor the water off when you need the mud again and don’t mix it in, the mud does not get significantly diluted. This would probably work well with typical wood filler tub for storage as long as you do not disturb it much. I do this to a lesser degree with dri-dex and acrylic filler as they will dilute; but a few little puddles on top in the tub will keep it moist without diluting it.
I usually buy wood filler by the small tube and still never finish those things before they have dried out too much to squeeze out properly. Wood putty I will sometimes soften with a bit of air tool oil if it has dried out / hardened too much.