The following question is considered for a smoothly parameterized family of control systems : Does there exist a smooth family of transformations (feedback only, feedback plus diffeomorphism, or feedback plus diffeomorphism with some restrictions on the parameter dependence of the diffeomorphism, called "matching conditions") changing the systems of the family into a single one? Some abstract necessary and sufficient conditions are given, under which an explicit construction of the transformation is proposed. Both local and global results are obtained. No constant rank assumption is needed for the general conditions, but they can be translated more explicitly under such assumptions.