It is known that monotone planar boolean circuits have a NC circuit value problem (in fact, much more is known).
What about non-boolean monotone planar circuits?
Precisely, take $Q=\{0,...,n-1\}$ and any family of monotone functions $f_i:Q^2\rightarrow Q$ and consider planar circuits where gates are chosen among the $f_i$. What is known for such circuits?