This is my current inventory of (fingerless) cycling gloves. With the possible exception of the low left pair, I’m pretty sure the Izumis, the Bianchis and the Specializeds have leather palms. And they’ve all been washed regularly w/o apparent damage.
The palms are leather. It's hard for me to see how running them through the wash will help.
Ever noticed how a towel will feel if you keep using it?
It's getting wet then drying them out that makes them so hard and stiff.
Sweat is more than water. There’s salt and all sorts of grime being worked into the leather, binding up the fibers. This is what causes most of the stiffness. Heck, if I don’t rinse my helmet straps regularly they’ll crinkle and creak when I put the helmet on. And there sure isn’t any leather in those.
Would you put leather shoes in the washer/dryer?
Depends on the shoe. Something like a Stan Smith - sure. Sneakers with leather trim - yes. A more ”formal” black/brown all leather shoe - probably not.
Look, this is advice freely given.
You are equally free to ignore it.
But the gloves are near useless to you now, what have you got to lose?
Or start softly if you wish.
After next ride, keep gloves on and proceed as if to wash your hands. Be generous with the soap. Keep washing until the water runs off clean.
Shape them, dry them at room temperature. See how they feel.