I was trying to give better clothes to a beggar and so I dropped the item beside him. He picked it up and asked if he could take it. I said yes. But he never equipped it.

I re-loaded to try again but when I dropped the item he didn't pick it up.

Far as I can see, NPCs can scramble or fight for items, but they never equip them. They just keep them around in the inventory for a bit.

Ive never seen them actually equip anything I drop. Theyll pick it up, then its gone...so I stopped dropping it in the end.

Only things I've ever seen NPCs equip are dropped weapons on roads - specifically Forsworn Swords being wielded by non-Forsworn NPCs.

They will with weapons. If you zone out and back in it should be equipped. I've reverse pickpocketed better weapons on to numerous NPCs and they all equip them. I dropped a Legendary Orcish Axe in Whiterun and watched for a while, but the NPCs just kicked it around. I left and when I came back a few game days or weeks later the beggar was wearing it and still is.

With clothes I've had no luck even when I strip off their body piece or give them actual armor.