After asking this question about signs in the Morse complex, I realised that my confusion is really about how tangent spaces to different (un)stable manifolds are related. So suppose we have a Morse function $f$ on a Riemannian manifold $M$, satisfying the Morse-Smale condition, with critical points at $x$ and $y$. Letting $W^s$ and $W^u$ denote stable and unstable manifolds respectively, suppose $W^u(x) \cap W^s(y)$ is non-empty, and pick a component $\gamma$. I would like to know the following:
Can we make sense of the limit of $TW^u(x)$ and $T\gamma$ in $T_yM$?
If so, is it true that $\lim_{p \rightarrow y} T_pW^u(x) = T_yW^u(y) \oplus \lim_{p \rightarrow y} T_p\gamma$?
These seem intuitively plausible, and would resolve my sign confusion, so I would be very grateful of either a reference or an explanation of how this works (or not!).
Added later: A statement of this sort seems to be implicit in Proposition 3.2 of Michael Hutchings's Morse homology notes, but without proof or reference.
Just parallel-transport along the gradient flow lines. You know that $\langle \mathrm{grad}_xf, v\rangle = (df)_x(v)$ for any $v \in T_xM$, and hence the sign of your inner product with the gradient picks out whether you're in the stable or unstable manifolds. Now recall that the inner product of tangent vectors is preserved under parallel transport.Maybe you also need that gradient flow lines are geodesics, too. I'm pretty sure this is true for Morse functions; see this MO question and realize that the only bad stuff can happen at critical points (which we don't care about in this setup).