I don't.  I just start throwing.  I don't start out at full strength, I work my way up in three or four throws.  Or a few more if the weather is cold.

I think that no warm up is a major advantage for a sling, you don't have to warm up to get pretty good power behind the throw.  That could have been of major importance way back when, after being crouched in a blind for a couple of hours waiting for waterfowl to come in.  If you tried to throw by hand, you will be stiff and not get a very strong throw.  With a mid length sling even a 3/4 throw is still way more powerful than most can throw by hand until they have warmed up a good bit.

I've a cranky rotater cuff. I have to warm up or itwill punish me later in the day after I cool down. Only a couple of minuets are needed and then I'm good to go.

I do, but not always, and when i do, it's just like a minute long hand rotation,and some sling throwing motion-imitations
For me, i think a two minute warmup would be plenty,at least for me.
Also,often beforehand slinging i'm doing some physical work,so i'm already,somewhat warmed up.

Only when I warm up, when I tried to throw 300m over the lake.

   I don't but I really should.  I warm up for everything else I do.  At my age (just turned sixty.. ugh!) I need to start warming up before I sling.

I rotate my wrists a bit, then do a couple of throws with the sling empty. My first few shots are usually pretty bad, no point in having to retrieve the ammo an extra time.

I don't ever sling at full power, otherwise I'd do a more decent warmup.