  We study lattice Miura transformations for the Toda and Volterra lattices,
relativistic Toda and Volterra lattices, and their modifications. In
particular, we give three successive modifications for the Toda lattice, two
for the Volterra lattice and for the relativistic Toda lattice, and one for the
relativistic Volterra lattice. We discuss Poisson properties of the Miura
transformations, their permutability properties, and their role as localizing
changes of variables in the theory of integrable discretizations.
