Christmas Trinity: Only the Son is incarnate, but the incarnation is the work of the whole Trinity. You can see why a distinction is helpful here: to recognize the undivided work of God toward us, but to specify the Son's incarnation exclusively.

Luther loved to use a homey image for this (one which he attributed to Bonaventure). The incarnation is like three girls putting a garment onto one of them: all three put it on, but only one has it put onto her.

Is it possible to be more precise? Well, although the work is undivided, the distinct persons are evident in the incarnation in a way that corresponds to their order of existence within the eternal relations: the Father unbegotten, the Son begotten, the Spirit proceeding.

Let's take the Father first: Just as he is the eternal origin of the Son, he is source of the Son's sending. The Father's sending of the Son is an extension of the Son's eternal begetting or generation. A real trinitarian mission makes known the eternal procession behind it.

This is why the Bible declares that the Father sends the Son: because this sending extends, or makes present among us, an eternal relation. Less specifically, we might say the Trinity sends the Son (or even that the Son sends himself!), but that's loose talk.