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

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

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

Is $\mathcal{E}^{M}_{\rm p}({S}_0, \mathbb{D}^n)$ 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 ${N} = \iota({S}_0) \cap {M}$ consists of those submanifolds which are "close" to ${N}$. 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 ${M}$ that is even diffeomorphic to ${N}$, let alone close.