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@Michael: I am trying to write down the details. One problem is that, on the critical submanifold $S$, there might be a codimension 1 submanifold which corresponds to those DEGENERATE critical points of $f(\cdot, y)$. When crossing these walls, the Morse index may jump. So when we try to construct the limit object for a given sequence, we try to find trajectories of $f(\cdot, y)$ from a component of $S$ to another component(by induction), and hope the induction may stop at finite times. But because of the jumping of indices, I have some trouble on deducing the finiteness.
Tameness is somewhat an open condition which is easier to handle. On the other hand, $J$-holomorphic curves for compatible $J$ are minimal surfaces, which are nicer. But anyway, as Michael said in the following answer, as far as I know, there's no much difference between the two choices.
