All paths of less than a given length in a directed graph between couple of nodes
If I am not missing something about your question, you can apply the standard trick for reducing the question about multiple sources and destinations to the case of single source and single destination: 


  
  Add two new nodes $s$ and $t$,  
  Connect $s$ to all sources,  
  Connect all destinations to $t$.
  


After this reduction we are in the standard question setting 
with two extra vertices $s$ and $t$ and 
we want to find walks from $s$ to $t$ of length at most $k+2$.