Has this formulation of pursuit evasion been researched? Similar to Helicopter Cops and Robbers Game
My friend Bastian and I devoted our bachelor's thesis to researching this problem. So I guess the problem has been researched (at least a little bit) now which answers my original question. Here is a link to the final result: The Monk Problem. 

Results include: 


New kind of recursive system (EL-systems) which is used as a verifier
Connection between search number and strongly connected components
Greedy heuristics based on graph decomposition


The complexity class still remains unknown.