Enumerating N-Dimensional Vectors
Pyth, 15 12 bytes

ms+0_%Q&gt;_zdQ


Test suite

My transformation is similar to one of xnor's, but in base 10. It works by unzipping the input into k separate numbers:

n = 21003034
k = 3

21003034
 1  3  4    134
2  0  3     203
  0  0        0


The numbers are ordered in decreasing position of the rightmost digit, so that all orderings of any group of numbers are possible.

The way the code works is we reverse the input, then slice off the last 0, 1, ... k-1 digits, then take every kth digit, reverse again, stick a 0 at the begining, and convert to int.