  The article is devoted to the investigation of the Noether currents and
integrals of motion in the special subclass of theories with higher field
derivatives -- theories under differential field transformations in action
(DFTA). Under fairly general assertions, we derive a simple representation for
the integrals as sums of "old" integrals of motion, terms, that vanishes on the
"old" equations of motion and surface integrals. We show that for some
cosmological theories of that kind with high order derivatives the last
contribution can violate gauge invariance, specifically, diffeomorphisms. Then
we investigate ambiguity of the Noether procedure for higher derivative
theories and fix it in the way the problem resolves in quite general setting.
The obtained results are discussed for the several extensions of the General
Relativity, which are obtained by DFTA, in particular, for mimetic, disformal
and Regge-Teitelboim theories of gravity.
