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The impact of electron-capture (EC) cross sections for neutron-rich nuclei on the dynamics of core collapse during infall and early post-bounce is studied by performing spherically symmetric simulations in general relativity using a multigroup scheme for neutrino transport and full nuclear distributions in extended nuclear statistical equilibrium models. We thereby vary the prescription for EC rates on individual nuclei, the nuclear interaction for the equation of state, the mass model for the nuclear statistical equilibrium distribution, and the progenitor model. In agreement with previous works, we show that the individual EC rates are the most important source of uncertainty in the simulations, while the other inputs only marginally influence the results. A recently proposed analytic formula to extrapolate microscopic results on stable nuclei for EC rates to the high densities and temperatures and the neutron-rich region, with a functional form motivated by nuclear-structure data and parameters fitted from large scale shell-model calculations, is shown to lead to a sizable (16%) reduction of the electron fraction at bounce compared to more primitive prescriptions for the rates, leading to smaller inner core masses and slower shock propagation. We show that the EC process involves -170 different nuclear species around 86Kr, mainly in the N=50 shell closure region, and establish a list of the most important nuclei to be studied in order to constrain the global rates. [ABSTRACT FROM AUTHOR]