Let $\mathbb{R}\ni t\mapsto g_t$ be a continuous path (in the $C^{\infty}$-topology of $\Gamma(\text{Sym}^2(T^*M))$) of smooth metrics on a compact spin manifold $M$. For each $t\in \mathbb{R}$, we can define the Dirac operator $D_t$ associated to the metric $g_t$ over a spinor bundle $S_t\to M$, in the usual way.
I'm trying to understand under which conditions one can perceive these Dirac operators as defined on the same spinor bundle $S$, with the final goal of showing that the map $t\mapsto D_t \in \mathcal{B}(W^1(S),L^2(S))$ is norm-continuous, where $W^1(S)$ denotes the first Sobolev space. I already know that for two conformally related metrics one can find an isometry between their spinor bundles, so this would definitely be enough, but too restrictive. I can see on Dependence of spinor bundle on choice of metric that the answer seems to be positive if one picks two metric $g_0$ and $g_1$ and looks at the line segment along them, but I cannot seem to be able to prove it.
Any help on that would be appreciated!