  We study some constructions on distributions in a uniform $p$-adic context,
and also in large positive characteristic, using model theoretic methods. We
introduce a class of distributions which we call distributions of ${\mathscr
C}^{\mathrm{exp}}$-class and which is based on the notion of ${\mathscr
C}^{\mathrm{exp}}$-class functions from [6]. This class of distributions is
stable under Fourier transformation and has various forms of uniform behavior
across non-archimedean local fields. We study wave front sets, pull-backs and
push-forwards of distributions of this class. In particular we show that the
wave front set is always equal to the complement of the zero locus of a
${\mathscr C}^{\mathrm{exp}}$-class function. We first revise and generalize
some of the results of Heifetz that he developed in the $p$-adic context by
analogy to results about real wave front sets by H\"ormander. In the final
section, we study sizes of neighborhoods of local constancy of Schwartz-Bruhat
functions and their push forwards in relation to discriminants.
