Let $M^r$ and $N^s$ be smooth submanifolds of the smooth manifold $V^{r+s}$ and $M,N,V$ compact, connected and orientable. The Thom-Isomorphism together with the Tubular Neighbourhood Theorem gives me a natural isomorphism $$\psi\ \colon \operatorname{H}_0(N)\rightarrow \operatorname{H}_r(V,V\backslash N)$$ Let now $\alpha$ be the canonical generator of $\operatorname{H}_0(N)\cong\mathbb{Z}$. One can proove that the orientation generator $[M]\in\operatorname{H}_r(M)$ is mapped to $(N\cdot M)\psi(\alpha)$ under the map $$\operatorname{H}_r(M)\rightarrow \operatorname{H}_r(V,V\backslash N)$$ induced by inclusion, where $N\cdot M$ denotes the intersection number. This shows, that one does not ned transversality for defining the intersection number and the intersection number does not change under deformations.
I am looking for a relative version of this lemma, that means I have compact connected, orientable manifold with boundary $W$ and compact, connected, orientable submanifolds $D$, $E$ with $\partial D\subset\partial W,\partial E\subset\partial W$ and I want to express the intersection number of $D$ and $E$ homologically like I did before without boundary.