I am attempting to understand and use the degeneration formula for GW invariants as stated in 'Gromov-Witten theory and Donaldson-Thomas theory II' (Maulik, Nekrasov, Okounkov, Pandharipande). The formula is found in section 3.4 of the aforementioned paper, but I will recall it here.

Let $\lambda:\chi\to C$ be a nonsingular 4-fold fibered over a nonsingular irreducible curve $C$ (I'm perfectly happy letting C = $\mathbb P^1$.) Let $X$ be a nonsingular fiber of $\lambda$ and $X_1\cup_S X_2$ be a reducible special fiber, consisting of two nonsingular 3-folds intersecting transversely along the the nonsingular surface (the relative divisor) $S$. The degeneration formula for the absolute invariants of $X$ in terms of the relative invariants of $X_1/S$ and $X_2/S$, is given as follows:

$$ Z'( X\ \left | \right. \prod _{i=1}^r \tau_0(\gamma_{l_i})_\beta = \sum Z'(\frac{X_1}{S}|\prod_{i\in P_1} \tau_0(\gamma_{l_i}))_{\beta_1,\eta} \xi(\eta) u^{2l(\eta)} Z'(\frac{X_2}{S}|\prod_{i\in P_2}\tau_0(\gamma_{l_i})_{\beta_2, \eta^{V}}$$.

In the above formula, $\eta$ is a weighted cohomology partition, and $\xi(\eta) = \prod_i \eta_i|Aut(\eta_i)$. $Z'$ is of course the GW partition function.

My Question: MNOP state that the sum is taken over curve splittings $\beta = \beta_1+\beta_2$. However, $\beta$ is a class on the nonsingular fiber $X$ and $\beta_1$ and $\beta_2$ are classes on the two components of the special fiber. I have tried to look at J. Li's paper that introduced the degeneration formula but I am unable to reconcile the two. The manner in which it is stated in MNOP II seems to indicate that the $\beta_i$ are classes on the components of the special fiber, but in that case I am not certain I understand how $\beta$ splits into these pieces. Are these classes implicitly being pulled back to classes on $X$ in some natural way?

Thanks in advance.