  We discuss field theories appearing as a result of applying field
transformations with derivatives (differential field transformations, DFT) to a
known theory. We begin with some simple examples of DFTs to see the basic
properties of the procedure. In this process the dynamics of the theory might
either change or conserve. After that we concentrate on the theories of gravity
which appear as a result of various DFT applied to general relativity, namely
the mimetic gravity and Regge-Teitelboim embedding theory. We review main
results related to the extension of dynamics in these theories, as well as the
possibility to write down the action of a theory after DFT as the action of the
original theory before DFT plus an additional term. Such a term usually
contains some constraints with Lagrange multipliers and can be interpreted as
an action of additional matter, which might be of use in cosmological
applications, e.g. for the explanation of the effects of dark matter.
