Has there been a study of circuits operating on arrays? Much ink has been spilled studying the theory surrounding computation by combinatorial circuits operating on bits or boolean values - with AND, OR and NOT gates (as those are enough to implement any boolean function). One can consider families of these for different input sizes as opposed to general Turing machines, and consider the languages they accept or functions they compute (if they have multiple outputs); one can reason regarding their size and depth as a function of the input length; define the classes of languages acceptable by them with certain restrictions; study their composition; etc.

I've also been introduced to the Blum-Shub-Smale computational model, where the constituent operations of a program have elements of a field (e.g. the Rationals or the Reals) as inputs and outputs; and such a machine can be likened to a circuit if it has no loops or similar control structures. It's been the subject of much less theoretical literature, but I can still find tens of thousands of papers referring to the "BSS Model". And there are arithmetic circuits (with only + and * gates) which I vaguely recall from my Complexity Theory class as an undergrad (proofs about #P and PSPACE where boolean circuits get "arithmetized" -  AND into * and OR into +, with the inputs still being boolean).

But - what about circuits which operate on more complex units of data? That is, arrays of unbounded finite length - with the individual elements either all of some single type or from some vocabulary of types? Obviously, you might want to constrain the nodes in such circuits, as otherwise they would just be Turing machine or circuit families in themselves, but this is still a relevant kind of automaton to study these days - with programming models such as CUDA and OpenCL which apply these kinds of operations on very large amounts of data, controlled at the level of the entire kernels and their dependencies.

So - have circuits-operating-on-array seen any theoretical study? I've failed to find any such work so far.

Note: RAM/register machines in which registers or individual memory cells contain arrays would be a next best thing.
Not really an answer, but would've been too long a comment. The result of Ben-Or and Cleve that arithmetic formulas can be simulated by branching programs of width 3, works over arbitrary non-commutative rings (in particular over the matrix ring $M_n(F)$). A nice consequence of this is that $\# NC^1$ circuits over $M_n(\mathbb{Z})$ captures ${\text{GapL}}$ over $\mathbb{Z}$ (for which the Determinant is a hard function). See for example, the paper on catalytic space by Buhrman et al. for an interesting generalization of the work of Ben-Or and Cleve. There is a result of Capelli, Durand and Mengel which gives a characterization of the arithmetic complexity class VP as equivalent to formulas whose nodes compute tensor contractions (note that here you even lose associativity).