Let $\mathscr{S}_0$ be a smooth compact $k$-dimensional manifold with boundary and $\mathcal{E}_{\rm p}(\mathscr{S}_0, \mathbb{D}^n)$ be the space of smooth proper embeddings into the unit disk in $\mathbb{R}^n$ with the weak topology, as defined in Hirsch. Let $\mathscr{M} \subseteq \mathbb{D}^n$ be a smooth compact $n$-dimensional submanifold with boundary.

Let $\mathcal{C}_k(\mathscr{M})$ be the space of $k$-dimensional properly embedded submanifolds of $\mathscr{M}$ with boundary, that is $$ \mathcal{C}_k(\mathscr{M}) = \coprod_{\mathscr{S}} \mathcal{E}_{\rm p}(\mathscr{S}, \mathscr{M})/\operatorname{Diff}(\mathscr{S}) $$ where the union runs over each diffeomorphism type. Define $$\mathcal{E}^{\mathscr{M}}_{\rm p}(\mathscr{S}_0, \mathbb{D}^n) = \{\iota \colon \mathscr{S}_0 \to \mathbb{D}^n \mid \iota \text{ is transverse to } \partial\mathscr{M}\}.$$

If an embedding is transverse then $\iota(\mathscr{S}_0) \cap \mathscr{M}$ is a submanifold of $\mathscr{M}$ with boundary. Hence, there is an intersection map $\mathcal{E}_{\rm p}(\mathscr{S}_0, \mathbb{D}^n) \to \mathcal{C}_k(\mathscr{M})$.

Is $\mathcal{E}^\mathscr{M}_{\rm p}(\mathscr{S}_0, \mathbb{D}^n)$ open, and is the intersection map continuous?

I believe the second question may be settled using an argument along the following lines. For every transverse embedding $\iota$, a basic open set $\mathcal{U}$ containing the intersection $\mathscr{N} = \iota(\mathscr{S}_0) \cap \mathscr{M}$ consists of those submanifolds which are "close" to $\mathscr{N}$. But the embeddings into $\mathbb{D}^n$ which are just as close to $\iota$ should map into $\mathcal{U}$.

However, it is not even clear how to show that a nearby embedding should have an intersection with $\mathscr{M}$ that is even diffeomorphic to $\mathscr{N}$, let alone close.