  We study upper bounds, approximations, and limits for functions of motivic
exponential class, uniformly in non-Archimedean local fields whose
characteristic is $0$ or sufficiently large. Our results together form a
flexible framework for doing analysis over local fields in a field-independent
way. As corollaries, we obtain many new transfer principles, for example, for
local constancy, continuity, and existence of various kinds of limits.
Moreover, we show that the Fourier transform of an $L^2$-function of motivic
exponential class is again of motivic exponential class. As an application in
the realm of representation theory, we prove uniform bounds for the normalized
by the discriminant Fourier transforms of orbital integrals on connected
reductive $p$-adic groups.
