Well, compare TG ( Transformational Grammar) and GPSG (Generalized Phrase Structure Grammar). TG allows the formulation of rules that perform long distance movement, using the variables of a transformation to stand for arbitrarily long strings of parts of a "proper analysis". For your example, a variable X of the WH movement transformation represents "John Past say that Mary will buy" (labeled brackets omitted), which is the material across which the "what" must move in order to get to the beginning of the sentence. The length of such a string is in principle unbounded, which makes this use of a variable "crucial".
At one time, it was thought that this was the killer argument for TG as against CFG (Context Free Grammar). For any given instance of WH movement, we can allow the movement or not, by fiddling with the details of the phrase structure rules of a CFG, but describing constraints on WH movement in a general way is impossible, simply because there would be an unbounded number of phrase structure rules to modify, but in principle a CFG can have only a finite, fixed number of rules.
This was the significance of the original title of Haj Ross's MIT dissertation, Constraints on Variables in Syntax.
However, if there are actually no rules in natural languages that perform such unbounded movements, then we perhaps no longer have this difficulty of the Ross constraints to deal with. Perhaps things move around in small steps, such as proposed in the successive cyclic proposal, where there are no unbounded movements. The implication is that each step can be constrained in some appropriate way, and so there will be many different ways of violating a constraint on WH movement, for instance, because movement within each cyclic domain is separately constrained.
It was Gerald Gazdar who described how to do this, by using "slash" features in a PSG to describe the distribution of gaps on a constituent-by-constituent basis. For instance, if it is possible for the CP after "say" to contain a gapped NP, we describe that by giving a phrase structure rule
VP/NP -> say CP/NP
which says that a possible VP with a NP which has been moved away can have "say" followed by a CP from which a NP has been removed. And then, we need also to say that a CP with a NP gap may have the form
CP/NP -> that S/NP
which is "that" followed by an S from which a NP has been removed.
In this way, movement is resolved into small steps, each of which can be described with a phrase structure rule (when the movement is actually possible). There is no unbounded movement possible, but we can hope that none is necessary.
One could say, then, that Chomsky's proposal makes use of Gazdar's idea, without giving credit, and without pointing out that the general form of the proposal apparently makes transformations altogether unnecessary.