We consider estimation and inference of a subvector of parameters that are defined through local moment restrictions. The framework is useful for a number of econometric applications including those in policy evaluation based on discontinuities or kinks and in real-time financial risk management. We aim to provide approaches to inference that are generic (without requiring case-by-case standard error analysis) and are robust when regularity assumptions fail. These irregularities include non-differentiability, non-negligible bias and weak identification. We focus on the QLR criterion-function-based (in particular, empirical likelihood-based) inference, and establish conditions under which the test statistic has a pivotal asymptotic distribution. Confidence sets can be obtained by inverting the test. In the key step of eliminating nuisance parameters in the criterion function, we consider that based on concentration and Laplace-type plug-in estimation. The former is natural, and the latter does not require optimization and can be computationally attractive in applications using simulations. We provide the asymptotic analysis under the null and local/non-local alternatives, and illustrate the high-levels assumptions with several examples. Simulations and an empirical application illustrate the finite-sample performance.