A few years ago I was wondering about (kind of) the same question and could not really find a satisfactory introduction. I settled for introductions to the general theory of foliations. I suppose you know references for that.

There are a few papers on foliations in algebraic geometry, especially in characteristic $p$. I understand that that is not what you are asking, but perhaps algebraic ideas might give you something in the Kahler case.

In particular, Miyaoka's paper,
MR927960 (89e:14011) 14E05 (14D99 14F10 14J40)
Miyaoka, Yoichi Deformations of a morphism along a foliation and applications. Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 245–268, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987.
is probably basic.

In general, it seems to me, that having an algebraic foliation is a very strong property. For instance, Kebekus-Solá Conde-Toma show that with some additional positivity properties an algebraic foliation implies very strong restrictions on the underlying manifold.

Again, I understand that this is not what you are asking for, but perhaps the references in this latter paper give you something to start and you might find that elusive introduction. If you do, please let me know. :)