Questions like this are so broad that the best response would be "read chapter X from book ABC" or "read the review article by XYZ". Unfortunately, I know of no good references for magnetostriction...so I'll try a rough sketch.
From my very basic (virtually layman) understanding of magnetostriction, I do not see that there should be a difference in the underlying principle on the basis of whether the material is ferromagnetic or antiferromagnetic. I have not come across a very good (and complete) explanation for magnetostriction anywhere, that gives an intuitive understanding of the phenomenon. So, what I think I understand comes essentially from keywords picked up from sources describing calculations of the various components of the magnetostrain tensor, and what-not.
Hence, at best, I can talk about the origin of magnetostriction in the general sense. I'm not sure what level of understanding you seek (are you a grad student specializing in magnetostriction calculations for specific AFM compounds, or are you in college, etc ?), so I'll assume very basic knowledge. Here's my attempt at providing a semiclassical feel for the principle behind magnetostriction...
The lattice constants of an insulating crystal are determined by the various interactions between the atomic nucleii and the electrons (and take on those values that minimize the free energy comprising of all these interaction terms). To a reasonable approximation, the atoms may be treated as ions with valence electrons. Another not terrible approximation is to consider only the interactions between nearest neighbors (the farther away you go, the weaker the interaction). So, in short, one of the things that determines the lattice parameters is the interactions between the valence electrons (of any given atom) and the neighboring ions.
One of the outcomes of building a crystal out of individual atoms is that the valence electron cloud (probability density |\psi | ^2) loses spherical symmetry, and now has a shape that minimizes the energy (a shape that reflects the crystal structure). The angular momentum of the valence electrons (which depends on the shape of this cloud) is thus a reflection of the crystal geometry. Since the magnetic moment (per ion) is a number proportional to the angular momentum of the valence electrons (classical equivalent : current loop has a magnetic moment; m = IA = (dq/dt)A = (e/T)A = ev(pi*r*r)/(2*pi*r) = (e/2)*r*v = (e/2m)*L, where L = angular mom. = m*v*r), this too depends on the crystal parameters. (Actually, in most materials, the total magnetic moment has a large component arising from the intrinsic electron spin, but this is also affected, due to spin-orbit coupling, which essentially makes the electron spin want to line up parallel to the orbital angular momentum). So, it would seem reasonable that altering the lattice parameters - which is the same as introducing a strain in the crystal - could change the way the magnetic moments want to point (by affecting the shapes of the electron clouds).
This is basically the inverse effect of magnetostriction (aka the Villari Effect), and says that applying stress to a magnetic material can change its magnetization. Reversing this line of thought explains magnetostriction, but needs to be done with a little care.
In the absence of an applied field, the atomic spins are lined up in some arbitrary (actually, they tend to orient along easy axes/planes, but a discussion of magnetic anisotropy is worthy of a whole thread by itself) direction (within a domain). Applying a magnetic field causes the spins to want to line up along the field. This causes the valence orbitals to distort so as to make the orbital moment line up parallel to the spins (and the applied field). The distortion of the valence orbitals causes the lattice parameters to change, which is seen as an effective strain in the crystal.
