Diffeological spaces are examples of generalized smooth spaces, they give a complete and cocomplete category of spaces including smooth spaces as a full subcategory. A diffeological space is a set together with a diffeology: a collection of plots (maps from a numerical domain) which determines the smooth structure.
For these diffeological spaces you have the concept of inductions and subductions.
Let me focus on inductions they are morphisms $f:(X,D)\rightarrow (X',D')$ such that $f^*(D')=D$ ($D$ is the pull-back diffeology of $D'$ along $f$) and $f$ is injective.
For smooth manifolds with their induced diffeology, every induction is an immersion but an immersion is not necessarily an induction even it is injective. Rather an immersion is a local induction.
Thus at least for diffeological spaces if you want to generalize immersions you have to work locally (with a superset of every point).
You also have local inductions, subductions.
You can have a look at P. Iglesias-Zemmour's book "Diffeology" (published by the AMS).