"reins"

Horses have reins, kings have reigns.

"Handing the reins over" is a reference to handing horse reins over.

There's not really a clear 'next-in-line'. Kadyrov has multiple sons, most of them young, and his leadership is based very heavily on the ties and agreements made with other Chechen groups, not organized, formal state institutions as we would recognize them. When he dies, all of that is up in the air - even the most bootlicking of them will be seeking to renegotiate on more favorable terms - and throwing their weight behind whoever promises that. Chechnya is not a 'normal' Russian republic. It's very much a special case.