  We construct a special-purpose functional flow equation which facilitates
non-perturbative renormalization group (RG) studies on theory spaces involving
a large number of independent field components that are prohibitively
complicated using standard methods. Its main motivation are quantum gravity
theories in which the gravitational degrees of freedom are carried by a complex
system of tensor fields, a prime example being Einstein-Cartan theory, possibly
coupled to matter. We describe a sequence of approximation steps leading from
the functional RG equation of the Effective Average Action to the new flow
equation which, as a consequence, is no longer fully exact on the untruncated
theory space. However, it is by far more "user friendly" when it comes to
projecting the abstract equation on a concrete (truncated) theory space and
computing explicit beta-functions. The necessary amount of (tensor) algebra
reduces drastically, and the usually very hard problem of diagonalizing the
pertinent Hessian operator is sidestepped completely. In this paper we
demonstrate the reliability of the simplified equation by applying it to a
truncation of the Einstein-Cartan theory space. It is parametrized by a scale
dependent Holst action, depending on a O(4) spin-connection and the tetrad as
the independent field variables. We compute the resulting RG flow, focusing in
particular on the running of the Immirzi parameter, and compare it to the
results of an earlier computation where the exact equation had been applied to
the same truncation. We find consistency between the two approaches and provide
further evidence for the conjectured non-perturbative renormalizability
(asymptotic safety) of quantum Einstein-Cartan gravity. We also investigate a
duality symmetry relating small and large values of the Immirzi parameter which
is displayed by the beta-functions in absence of a cosmological constant.
