In the paper Tame flows I have investigated a special class of gradient like flows.
The Conley index theory of such flows is particularly easy to describe and leads to a result that generalizes the well known theorem in Mores theory stating that "crossing a critical point amounts to attaching a cell of appropriate dimension''. (Sec. 9 and 10 of the above paper).
In Sec 11 I show that an injective function $f$ from the faces of a simplicial complex to the reals naturally defines one such flow. Its stationary (or critical) points are the barycenters of the faces. In particular such a function leads to a homotopical reconstruction of the space different from the one given by the simplicial decompositions. The attaching spaces when crossing a critical point can be identified naturally with the unstable variety of that point which is a subcomplex that can be described explicitly in terms of the function $f$.
The Morse-Foreman functions have one appealing property. The Conley index of the barycenter of a Foreman non-critical face is homotopically trivial, while the Conley index of the barycenter of a Foreman critical face is of sphere of dimension equal to the dimension $k$ of the critical face. Crossing such a critical point corresponds to attaching a disk of dimension that be canonically identified with that face.
The precise details are in Sec 9-11 of the above paper. I want to mention one other thing. In Sec 11 I tried with modest success to address one limitation of Morse-Foreman theory, namely the scarcity of Morse-Forman functions. Usual Morse functions on smooth manifolds are "a dime a dozen" in the sense that generic smooth functions are Morse or better, yet a smooth function is almost surely Morse. In the discrete case, the probability that a random assignments of numbers to faces yields a discrete Morse function is very small, in fact exponentially small in the number of faces.
In Sec. 11 I describe a larger class of functions on the set of faces of a simplicial complex that contains the discrete Morse-Foreman functions and homotopically behave like Morse-Forman functions. Unfortunately, even this larger class is rather exponentially thin.