Moving window calculations arise throughout applied mathematics, computer science, and engineering. Familiar examples include moving sums, maxima, and filters, but the same basic problem also appears in more general windowed recurrences. In this talk, we show that a large class of such calculations admits a common algebraic formulation. In the associative setting, moving window calculations can be recast as exponentiation in a semidirect product, bringing efficient powering algorithms to bear. In more general settings, semi-associativity provides the right framework for representing compositions of updates and understanding when efficient computation is still possible. This point of view not only unifies a broad family of examples, but also leads naturally to compact vectorized implementations, making the methods well suited to modern GPU architectures. This is joint work with David Maslen.