  Miura-type transformations (MTs) are an essential tool in the theory of
integrable nonlinear partial differential and difference equations. We present
a geometric method to construct MTs for differential-difference (lattice)
equations from Darboux-Lax representations (DLRs) of such equations.
  The method is applicable to parameter-dependent DLRs satisfying certain
conditions. We construct MTs and modified lattice equations from invariants of
some Lie group actions on manifolds associated with such DLRs.
  Using this construction, from a given suitable DLR one can obtain many MTs of
different orders. The main idea behind this method is closely related to the
results of Drinfeld and Sokolov on MTs for the partial differential KdV
equation.
  Considered examples include the Volterra, Narita-Itoh-Bogoyavlensky, Toda,
and Adler-Postnikov lattices. Some of the constructed MTs and modified lattice
equations seem to be new.
