Local consistency as a reduction between constraint satisfaction problems

We study the use of local consistency methods as reductions between constraint satisfaction problems (CSPs), and promise version thereof, with the aim to classify these reductions in similar way as the algebraic approach classifies gadget reductions between CSPs. We classify a use of arc-consistency in this way, provide first steps into classification of general $k$-consistency, and ask whether every tractable finite template CSP is reducible by such a reduction to solving systems of affine Diophantine equations.