Are function spaces over a shrinking set vector bundles?

Question

I have function spaces $F(S_t) \subseteq \{f : S_t \to \mathbb{R}\}$ defined over shrinking sets $S_t\subset S_s$ for $t> s$. I have a trivial fiber bundle $[0,\infty) \times F(S_0)$ where the projection map is $\pi((t,f)) = t$. I now consider the subbundle where at each $t$ I instead take the space $F(S_t)$. In order for this to be a fiber bundle I need a homeomorphism $\varphi : \pi^{-1}((a,b)) \to (a,b) \times H$ that locally trivializes the bundle and the projection from $(a,b)\times H$ agrees with $\pi$. My issue is finding a homeomorphism $\varphi$. For example if I were to take $H = F(S_a)$ then I could extend any function $f \in F(S_t)$ for $t\in (a,b)$ to a function from $F(S_a)$ by setting it to zero outside of $S_t$. In that case $\varphi$ would be injective, but not surjective. On the other hand if I were to take $H = F(S_b)$ and map any function $f\in F(S_t)$ to $F(S_b)$ by its restriction to $S_b$ then the map is not injective. So my conclusion was that this is not a fiber bundle. Is there some other notion that generalizes fiber bundles that would allow for e.g. the case where $\varphi$ is not surjective? I am looking for some reference that formalizes the above in a similar way as is done for fiber bundles. This arose from looking at PDEs with a shrinking domain due to a contracting boundary.