  In 2006, Fukaya and Kato formulated a general conjecture about $p$-adic
$L$-functions for a large class of motives and derived a precise conjectural
interpolation formula from the ETNC. We study this conjectural $p$-adic
$L$-function in the setting of Hida families and compare it to the $p$-adic
$L$-function constructed by Kitagawa. A priori it is not clear whether the
interpolation formulas coincide, mainly because of the rather inexplicit
definition of periods (via comparison isomorphisms) that appear in the general
formula.
  Using $p$-adic Eichler-Shimura isomorphisms and their interpolation in Hida
families, we compare these periods to the expressions in Kitagawa's formula
coming from modular symbols. This allows in certain cases to modify Kitagawa's
construction such that it produces a $p$-adic $L$-function compatible with
Fukaya's and Kato's conjecture.
