Assume that $\mathcal{R}$ is a hyperbolic surface with $m$ geodesic boundary components and $n$ punctures. If $\mathcal{R}'$ is a closed subsurface of a hyperbolic surface $\mathcal{R}$, then there is an inclusion homomorphism between the mapping class groups:
$$\text{Mod}(\mathcal{R}')\longrightarrow \text{Mod}(\mathcal{R})$$
I am concerned with the situation where $\mathcal{R}''$ is a general subsurface of $\mathcal{R}$. Such a surface has some handles, a number of boundaries, and a number of punctures. The presence of punctures makes the surface non-closed assuming that the boundary curves belong to the boundaries of $\mathcal{R}''$. It is possible some of the boundaries and/or punctures of $\mathcal{R}''$ are also boundaries and/or punctures of $\mathcal{R}$, i.e :
$$\partial\mathcal{R}''\cap \partial\mathcal{R}\ne \emptyset$$
Here a mapping-class fixes the boundary but can permute the punctures.
There are two questions:
- Is there an inclusion homomorphism between the mapping-class groups in this case, i.e. does a homomorphism $\text{Mod}(\mathcal{R}'')\longrightarrow \text{Mod}(\mathcal{R})$ exist?
- If yes, when does such a homomorphism is injective? In particular, when $\text{Mod}(\mathcal{R}'')$ is a subgroup of $\text{Mod}(\mathcal{R})$?
The case where the mapping-class group fixes the puncture and all of the boundaries of $\mathcal{R}''$ belong to the interior of $\mathcal{R}$ is treated in Geometric Subgroups of Mapping Class Groups.
Your first question has a simple negative answer. Suppose that $\mathcal{R}$ is a closed oriented surface and $\mathcal{R}'' = \mathcal{R}-p$ for some point $p$ in the interior of $\mathcal{R}$. Then a canonical homomorphism $$\text{Mod}(\mathcal{R}'') \mapsto \text{Mod}(\mathcal{R}) $$ is defined, because every homeomorphism of $\mathcal{R}''$ has a unique extension to a homeomorphism of $\mathcal{R}$. But the kernel of this homomorphism is isomorphic to the fundamental group of $\mathcal{R}$, according to the Birman short exact sequence, hence the homomorphism is not injective.