Jim Keltner says The Stones were Charlie's band. ;-)
Charlie was well known for creating the "feel" of The Stones; he played the snare drum back beat real "late" (far back in the pocket, in drumming lingo). A lot of Rockers who discovered Country music in the late-60's/early-70's (Gram Parsons made Country hip to some), as well as The Stones feeble attempts to play it (Keith came under the influence of Gram), want their drummer to play their Country-Rock ala Charlie, with that late back beat.
I invested a lot of time and devoted a lot of work to be able to play "deep pocket" style---like Roger Hawkins of The Muscle Shoals Rhythm Section aka The Swampers (Aretha, Boz Scaggs, Joe Cocker, Paul Simon, The Staple Singers, Duane Allman, Albert King, Bob Seger, Etta James, Wilson Pickett, hundreds of others. One of my two favorite drummers), Jim Gordon (Derek & The Dominos, All Things Must Pass, Traffic, L.A. studios), Al Jackson (Booker T & The MG's), Ringo, and the late, great Levon Helm (my other favorite).
In the first decade of this century I did some gigs with Rick Shea and Jonny Kaplan, a couple of such guys, well-known in the L.A. musical community. After working so hard at being able to play in the center of the pocket (think of the pocket as a V-shaped hole. The closer you move to the center of the hole, the deeper the pocket), intentionally playing late was not fun. It was in fact painful to my musical sensibilities. But, as the saying goes, give them what they want. ;-)