C(9,4) / 64 = 126 / 1296 = 7 / 72
as each choice of four integers from the range 1 through 9 corresponds with exactly one set of rolls that satisfy the desired condition.
Consider the rolls outcomes labeled as a,b,c,d. The enhanced number sets as A,B,C,D, where A=a, B=b+1, C=c+2, D=d+3.
The set A=3,B=5,C=6,D=8 corresponds to rolls of 3,4,4 and 5, and since there is inequality guaranteed by the choice of four out of nine, there is only one set, that can be placed in order, and will be in one-to-one correspondence with the original set.
As alluded to in Ady Tzidon's comment, separate track can be kept of sequences ending in 1, 2, ..., 6 and added up, starting with a sequence of 1, then 2, then 3 leading up to 4:
1+1+1+1+1+1 = 6
1+2+3+4+5+6 = 21
1+3+6+10+15+21 = 56
1+4+10+20+35+56 = 126
where each n'th term is the sum of the first n terms in the row before.
Then, of course, the sum 126 is divided by 64.