Head versus Phrase
If the difference between head and phrasal movement does not reside in the movement module, the only alternative is to rely on phrase structure theory. There is indeed a primitive difference between heads and phrases, which we will capitalize on to derive the complementarity we need: in a nutshell, a head projects; a phrase is a projection. This primitive difference is enough to de rive in a principled way (1) the complementarity between head movement and phrasal movement, (2) the choice of phrasal movement in standard wh-constructions, and (3) the existence of minimally different wh constructions involving head movement.
But let us proceed step by step, trying to be really minimal. Suppose a probe a on a head A at the root attracts a goal b. Given that a feature cannot be merged, some extra material needs to be merged (call it B), and the offending feature is deleted (1). The operation Merge (A, B) is asymmetric in essence, so that given the configuration in (1) one of two things must happen: either A or B must project, yielding the two configurations (2) or (3). The element that projects is the head of the resulting phrase.
(1)

Let us combine these trivial assumptions with what we know about movement. Suppose a moved item and its copy must be the same with respect to their phrase structure status (the CUC). This means that when a head, which by definition projects, gets moved, it must project; but when a phrase, itself by definition a projection, gets moved, it does not project. As a result, whenever a feature moves as a head, all the features it is associated with (and notably the categorial feature) project. Therefore, head movement changes the feature composition of the target. When the very same feature moves as a phrase, this does not happen, and the target remains unchanged.
The economy condition on Internal Merge ensures that the two movements never overlap: if the grammar always copies enough material for convergence, it will select head movement unless convergence at LF (interpretation) chooses differently.
To see how this account works, let us look at (standard) wh constructions. In interrogatives, the complementizer head selecting the structure contains a feature, call it [wh], that needs to be checked, acting as a probe. The goal corresponds to a wh-element embedded in the clause, (a copy of) which needs to be merged in a local configuration with the probe. Since features cannot be merged, the minimal option is for the wh-feature attracted by C to pied-pipe the wh-word alone (head movement). But this minimal option does not yield a convergent derivation: moving the wh-feature as a head means projecting all the features associated with it, and notably its categorial feature (D). This would turn the interrogative clause into a complex DP as in (2), which is not interpretable as an interrogative clause at the interface. This is why the more costly derivation (3) is selected.
(2)

(3)

Summarizing so far, there is only one operation, Move, which is triggered by a feature and defined at that level, and which merges just enough material for convergence. Then, convergence at LF decides whether ‘‘enough material’’ is a head (which retains its projection property throughout the derivation, given the CUC) or a phrase (which remains a projection throughout the derivation). In standard wh-constructions, such as interrogatives, LF convergence selects the less minimal option, that of moving the entire phrase, preserving the simple CP categorial status of the clause.