Left Center Right (LCR) code golf
Emacs Lisp, 279 bytes

(defmacro n(i)`(incf(nth ,i c)))
(defun f(p s)(g(let((a'(0)))(dotimes(i p)(push s a))(princ a))0 p))
(defun g(c v p)(dotimes(i(min(nth v c)3))(decf(nth v c))(case(random 6)(0(n(mod(1- v)p)))(1(n(mod(1+ v)p)))(2(n p))(t(n v))))(princ c)(or(eq(-sum c)(nth p c))(g c(mod(1+ v)p)p)))


Use this function as (f 4 3).

Better readable version:

(defmacro n (i) `(incf (nth ,i c)))

(defun f(p s)
  (g
   (let ((a '(0)))
     (dotimes (i p)
       (push s a))
     (princ a))
   0
   p))

(defun g (c v p)
  (dotimes (i (min (nth v c) 3))
    (decf (nth v c))
    (case (random 6)
      (0 (n (mod (1- v) p)))
      (1 (n (mod (1+ v) p)))
      (2 (n p))
      (t (n v))))
    (princ c)
    (or (eq (-sum c) (nth p c))
    (g c (mod (1+ v) p) p)))


Output example:

(3 3 3 3 0)(1 4 3 4 0)(2 2 4 4 0)(2 2 2 5 1)(4 2 2 3 1)(2 2 2 4 2)(2 1 3 4 2)(2 2 0 4 4)(2 2 0 4 4)(1 2 0 4 5)(2 1 0 4 5)(2 1 0 4 5)(2 1 1 3 5)(0 1 1 3 7)(1 0 1 3 7)(1 0 1 3 7)(1 0 3 1 7)(1 0 3 1 7)(1 0 3 1 7)(1 1 2 1 7)(1 1 3 0 7)(0 1 3 0 8)(1 0 3 0 8)(1 1 1 1 8)(1 1 2 0 8)(0 1 2 1 8)(0 1 2 1 8)(0 1 1 1 9)(0 1 1 1 9)(0 1 1 1 9)(0 1 1 1 9)(0 1 1 1 9)(0 1 1 0 10)(0 1 1 0 10)(0 0 1 0 11)(0 0 1 0 11)(0 0 1 0 11)(0 0 1 0 11)(0 0 1 0 11)(0 0 0 0 12)