There are three ways, as far as I can see, that you can interpret the fact that many of the clubs Allardyce has left have fallen on difficult times.
1. It's a coincidence - the least likely, in my opinion.
2. That the clubs that have recruited Allardyce in recent times had major structural problems before, during and after Allardyce's tenure
3. That Allardyce's approach to squad building is only sustainable in the short-term and, if this is not recognised quickly by the new incumbent, then this may have a detrimental effect on the medium term performances of the club
Occam's razor would suggest that it can't be all Allardyce's doing but, equally, that he is a common factor in an ongoing trend and his contribution cannot be discounted. Therefore, in my view, a combination of points 2 and 3 seems to be the answer that relies on the least number of assumptions.