What is the big version of NC?
I believe the class you are looking for is $PSPACE$. Suppose you have $exp(n^k) = 2^{O(n^k)}$ processors fitting the requirements:


  
  Every computer reads the polynomial number of polynomial sized messages it received in  the previous time step.
  Every computer runs some polytime computation that can depend on these messages.
  Every computer passes a message (of polylength) to each of its neighbours.
  


This can be modeled by having a circuit with $poly(n)$ layers, where each layer has $exp(n^k)$ "gates", and each "gate" does a polynomial time computation (satisfying part 2) with polynomial fan-in (satisfying part 1), and has polynomial fan-out (satisfying part 3).  

Since each gate computes a polynomial time function, they each can be replaced by a polynomial size circuit (with AND/OR/NOT) in the usual way. Note the polynomial fan-ins and fan-outs can be made to be 2, by only increasing the depth by a $O(\log n)$ factor. What remains is a $poly(n)$ depth uniform circuit with $exp(n^k)$ AND/OR/NOT gates. This is precisely alternating polynomial time, which is precisely $PSPACE$.