Change of order for regular chains in positive dimension

We discuss changing the variable order for a regular chain in positive dimension. This quite general question has applications going from implicitization problems to the symbolic resolution of some systems of differential algebraic equations. We propose a modular method, reducing the problem to computations in dimension zero and one. The problems raised by the choice of the specialization points and the lack of the (crucial) information of what are the free and algebraic variables for the new order are discussed. Strong (but not unusual) hypotheses for the initial regular chain are required; the main required subroutines are change of order in dimension zero and a formal Newton iteration.