Nonequilibrium conditions give rise to classes of universally evolving configurations of quantum many-body systems at nonthermal fixed points. While the fixed point and thus full scaling in space and time is generically reached at very long evolution times, we propose that systems can show prescaling much earlier in time, in particular, on experimentally accessible timescales. During the prescaling evolution, some well-measurable properties of spatial correlations already scale with the universal exponents of the fixed point while others still show scaling violations. Prescaling is characterized by the evolution obeying conservation laws associated with the remaining symmetry which also defines the universality class of the asymptotically reached nonthermal fixed point. Here, we consider N=3 species of spatially uniform three-dimensional Bose gases, with identical inter- and intraspecies interactions. During prescaling, the full U(N) symmetry of the model is broken to U(N-1) while the conserved transport, reflecting explicit and emerging symmetries, leads to the buildup of rescaling quasicondensate distributions.