Let $M$ be an $n$ dimensional smooth manifold and let $j: M \to \mathbb{R}^{m}$ be an embedding. Associated to this embedding we can form the "collapse map" which is a pointed map from a sphere to the Thom space of the normal bundle $S^{m}=(\mathbb{R}^m)^{+} \to Th(N_j)$ (which depends on the choice of tubular neighborhood). The homotopy class of the collpase map is however well defined as a function of the isotopy class of the embedding. The "collapse map" is, in laconic terms, a well defined function of sets:

$$\{{ \text{embeddings } M \hookrightarrow \mathbb{R}^m \}/\sim_{\text{isotopy}}} =: E \longrightarrow \coprod_{[j] \in E} [S^m,Th(N_j)]$$

This might be fine for the most part but it seems desirable (even from the POV of applications) to have a version of this construction which works for families.

Question: Is there a "natural" (in the sense that it does what you expect it to do for families of embeddings) map (/homotopy class of maps) of spaces:

$$Emb(M,\mathbb{R}^m) \to \coprod_{[j] \in \pi_0(Emb(M,\mathbb{R}^m))} Map_*(S^m,Th(N_j))$$

Which on connected components induces the collapse map construction above?