Dealing with missing data via parametric multiple imputation methods usually implies stating several strong assumptions both about the distribution of the data and about underlying regression relationships. If such parametric assumptions do not hold, the multiply imputed data are not appropriate and might produce inconsistent estimators and thus misleading results. In this paper, a fully nonparametric and a semiparametric imputation method are studied, both based on local resampling principles. It is shown that the final estimator, based on these local imputations, is consistent under fewer or no parametric assumptions. Asymptotic expressions for bias, variance and mean squared error are derived, showing the theoretical impact of the different smoothing parameters. Simulations illustrate the usefulness and applicability of the method.