I've cleaned my leather shammy with warm water and soap and a nylon suede brush and it's fine.

Use a very mild soap, like hand soap. Don't use dish soaps, laundry detergents, chemicals , alcohol or you'll remove the natural oils from the leather.

Leather shammys usually get stiff because they get too dry and loss of the natural oils in the leather.

I washed mine by hand with a hand soap and gently scrubbed it with the brush, then rinsed it thoroughly and squeezed/wrung it dry.

I then stretched it out and clipped it to a coat hanger and let it air dry. Don't put it in the sun, UV is bad for it.