This is a computerized solution. I encourage solvers to treat it as a sort of "answer key" (because computerized solutions are boring) and have broken it into multiple spoilers to better facilitate partial reveals.
Broken down by segments corresponding to justhalf's solution above, the number of moves in each section are:
Step 1:
Priest C makes 9 moves to unblock Priest A
Step 2:
Priests C and A make 10 moves to unblock priest D
Step 3 and 4:
Priests C, A, and D make 17 moves to unblock priest B
Step 5:
The four priests together make 8 moves to reach the summit.
For a total of
44 moves
The moves are
Step 1:
1: C<4 : Move Priest C to the top of domino 9
2: C : Move Priest C to the bottom of domino 9
3: C>1 : Move Priest C to the bottom of domino 10
4: C : Move Priest C to the top of domino 10
5: C>2 : Move Priest C to the top of domino 12
6: C<4 : Move Priest C to the top of domino 8
7: C : Move Priest C to the bottom of domino 8
8: C<5 : Move Priest C to the bottom of domino 3
9: C : Move Priest C to the top of domino 3
Step 2:
10: A>3 : Move Priest A to the top of domino 4
11: A : Move Priest A to the bottom of domino 4
12: A>1 : Move Priest A to the bottom of domino 5
13: C : Move Priest C to the bottom of domino 3
14: C>5 : Move Priest C to the bottom of domino 9
15: A>5 : Move Priest A to the bottom of domino 11
16: C>1 : Move Priest C to the bottom of domino 10
17: C : Move Priest C to the top of domino 10
18: C>2 : Move Priest C to the top of domino 12
19: C : Move Priest C to the bottom of domino 12
Steps 3 and 4:
20: D<1 : Move Priest D to the bottom of domino 10
21: C : Move Priest C to the top of domino 12
22: C<4 : Move Priest C to the top of domino 8
23: C : Move Priest C to the bottom of domino 8
24: C<5 : Move Priest C to the bottom of domino 3
25: C>1 : Move Priest C to the bottom of domino 4
26: C>1 : Move Priest C to the bottom of domino 5
27: D : Move Priest D to the top of domino 10
28: A<1 : Move Priest A to the bottom of domino 10
29: D>2 : Move Priest D to the top of domino 12
30: D<4 : Move Priest D to the top of domino 8
31: D : Move Priest D to the bottom of domino 8
32: A<5 : Move Priest A to the bottom of domino 3
33: D<4 : Move Priest D to the bottom of domino 2
34: A : Move Priest A to the top of domino 3
35: C<3 : Move Priest C to the bottom of domino 1
36: A : Move Priest A to the bottom of domino 3
Step 5:
37: B>1 : Move Priest B to the bottom of domino 4
38: C>3 : Move Priest C to the bottom of domino 7
39: C : Move Priest C to the top of domino 7
40: D>3 : Move Priest D to the bottom of domino 7
41: A : Move Priest A to the top of domino 3
42: A>3 : Move Priest A to the top of domino 6
43: D : Move Priest D to the bottom of domino 7
44: B>3 : Move Priest B to the bottom of domino 7
