  Zero forcing is a process on a graph in which the goal is to force all
vertices to become blue by applying a color change rule. Throttling minimizes
the sum of the number of vertices that are initially blue and the number of
time steps needed to color every vertex. We introduce a new universal
definition of throttling for variants of zero forcing and the study of
throttling for the minor monotone floor of zero forcing. We introduce the
technique of using a zero forcing process to extend a given graph. For standard
zero forcing and its floor, we use these extensions to characterize graphs with
throttling number $\leq t$ as certain minors of Cartesian products of complete
graphs and paths. We apply these characterizations to determine graphs with
extreme throttling numbers.
