Boolean circuits are non-uniform models of computation in that they require a different circuit for each length of input. The typical way of uniformizing a family of Boolean circuits is to define a Turing machine that can output, for some input length N, the correct Boolean circuit for that N.
However, I wonder if it is instead possible, if not a bit abstract, to instead look at finite Boolean circuits that "converge" to some well-defined infinite circuit in the large limit of N, which can handle any input size of arbitrarily large but finite length, and get an equivalent model of computation.
My intuition is that this is often the case, because it is quite common for Boolean circuits to be able to handle not only their own input size, but all smaller input sizes as well. For example, an adder that works on inputs of N bits can handle inputs of length < N by zero-padding the highest undefined bits; this is equivalent to setting the undefined bits to zero by default. So we can easily imagine this series of circuits converging to something like ripple adder extending infinitely to the left, and for which we consider only those inputs with finitely many nonzero bits.
What I am really trying to understand is the simplest way to extend the notion of Boolean circuit to get something that is Turing-complete. For example, we can give a finite state machine an infinite queue and make it Turing complete. By looking at infinite Boolean circuits that can handle arbitrarily large input with finite support, can we likewise obtain Turing-completeness, with an analogue of the halting problem, etc? If not, how could we do it?