Given any task $Π$, Brandt's speedup theorem (PODC 2019) provides a mechanical way to design another task~$Π'$ on the same input-set as $Π$ such that, for any $t\geq 1$, $Π$ is solvable in $t$ rounds if and only if $Π'$ is solvable in $t-1$ rounds. The theorem applies to the anonymous variant of the LOCAL model, in graphs with sufficiently large girth, and to locally checkable labeling (LCL) tasks. In this paper, using combinatorial topology applied to distributed computing, we dissect the construction in Brandt's speedup theorem for expressing it in the broader framework of round-based models supporting full information protocols, which includes models as different as wait-free shared-memory computing with iterated immediate snapshots, and synchronous failure-free network computing. In particular, we provide general definitions for notions such as local checkability and local independence, in our broader framework. In this way, we are able to identify the hypotheses on the computing model, and on the tasks, that are sufficient for Brandt's speedup theorem to apply. More precisely, we identify which hypotheses are sufficient for the each direction of the if-and-only-if condition. Interestingly, these hypotheses are of different natures. Our general approach enables to extend Brandt's speedup theorem from LOCAL to directed networks, to hypergraphs, to dynamic networks, and even to graphs including short cyclic dependencies between processes (i.e., the large girth condition is, to some extend, not necessary). The theorem can even be extended to shared-memory wait-free computing. In particular, we provide new impossibility proofs for consensus and perfect renaming in 2-process systems.