  We present an application to cosmological models in $f({\cal R})$ theories
within the Palatini formalism of a method that combines cosmography and the
explicit form of the field equations in the calculation of the redshift drift.
The method yields a sequence of constraint equations which lead to limits on
the parameter space of a given $f({\cal R})$-model. Two particular families of
$f({\cal R})$-cosmologies capable of describing the current dynamics of the
universe are explored here: (i) power law theories of the type $f({\cal
R})={\cal R}-\beta /{\cal R}^n$, and (ii) theories of the form $f({\cal
R})={\cal R}+\alpha \ln{{\cal R}} -\beta$. The constraints on $(n,\beta)$ and
$(\alpha,\beta)$, respectively, limit the values to intervals that are narrower
than the ones previously obtained. As a byproduct, we show that when applied to
General Relativity, the method yields values of the kinematic parameters with
much smaller errors that those obtained directly from observations.
