Let $\xi$ be a vector bundle over a closed manifold $X$. A framing (or trivialization) of $\xi$ is an isomorphism of bundles $\xi \to \epsilon^n$ where $\epsilon^n$ is an $n$-dimensional trivial bundle over $X$. A stable framing of $\xi$ is an isomorphism $\xi \oplus \epsilon^k \to \epsilon^{n+k}$. Typically, two framings are considered equivalent if they are homotopic through bundle isomorphisms and two stable framings are considered equivalent if some stabilization of them is homotopic through bundle isomorphisms.
I would like an example of a sequence bundles $\xi_k$ (ideally, all over the same base space) and pairs of trivializations $\phi_1, \phi_2 : \xi \oplus \epsilon^m \to \epsilon^n$ with the property that $\phi_1$ and $\phi_2$ are stably equivalent but $\phi_1$ and $\phi_2$ are not equivalent when just stabilized $k$ times.