The dimensional-reduction hypothesis is tested in theories with a first-order phase transition. Five-dimensional SU(2) lattice gauge theory has a confined phase which resembles the two-dimensional theory; across the phase transition this resemblance is lost and the string tension vanishes. A similar pattern is found for reduction from four-dimensional SO(3) to the two-dimensional SO(3) theory. An external random flux does not induce reduction where it is not spontaneously present.