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.
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