  In the 1970's, Serre exploited congruences between $q$-expansion coefficients
of Eisenstein series to produce $p$-adic families of Eisenstein series and, in
turn, $p$-adic zeta functions. Partly through integration with more recent
machinery, including Katz's approach to $p$-adic differential operators, his
strategy has influenced four decades of developments. Prior papers employing
Katz's and Serre's ideas exploiting differential operators and congruences to
produce families of automorphic forms rely crucially on $q$-expansions of
automorphic forms.
  The overarching goal of the present paper is to adapt the strategy to
automorphic forms on unitary groups, which lack $q$-expansions when the
signature is of the form $(a, b)$, $a\neq b$. In particular, this paper
completely removes the restrictions on the signature present in prior work. As
intermediate steps, we achieve two key objectives. First, partly by carefully
analyzing the action of the Young symmetrizer on Serre-Tate expansions, we
explicitly describe the action of differential operators on the Serre-Tate
expansions of automorphic forms on unitary groups of arbitrary signature. As a
direct consequence, for each unitary group, we obtain congruences and families
analogous to those studied by Katz and Serre. Second, via a novel lifting
argument, we construct a $p$-adic measure taking values in the space of
$p$-adic automorphic forms on unitary groups of any prescribed signature. We
relate the values of this measure to an explicit $p$-adic family of Eisenstein
series. One application of our results is to the recently completed
construction of $p$-adic $L$-functions for unitary groups by the first named
author, Harris, Li, and Skinner.
