Many optimization problems require the use of a local search to find a satisfying solution in a reasonable amount of time, even if the optimality is not guaranteed. Usually, local search algorithms operate in a search space which contains complete solutions (feasible or not) to the problem. In contrast, in Consistent Neighborhood Search (CNS), after each variable assignment, the conflicting variables are deleted to keep the partial solution feasible, and the search can stop when all the variables have a value. In this paper, we propose a generalized version of CNS, discuss its performance according to various criteria, and present successful adaptations of CNS to three types of satellite range scheduling problems. Such problems are motivated by applications encountered by the French National Space and Aeronautic Agencies and the US Air Force Satellite Control Network. The described numerical experiments will demonstrate that CNS is a powerful and flexible method, which can be easily combined with efficient ingredients.