I have a piece of juvenilia on this topic, considering the case of zero-dimensional submanifolds. See Section 9 of

O. Randal-Williams, Embedded cobordism categories and spaces of submanifolds, IMRN 3 (2011) 572-608.

It concerns how close the relationship is between

1. the cobordism category having oriented 0-manifolds inside $M$ as objects, and oriented 1-dimensional cobordisms in $M \times [0,t]$ as morphisms, and
2. the fundamental (topological) groupoid of McDuff's space of annihilating positive and negative particles in $M$,

is.

The main theorem in this direction is that while the categories are in no sense equivalent as topological categories (the circle as a cobordism $\emptyset \leadsto \emptyset$ is contractible as a loop in McDuff's space), they do nontheless have homotopy equivalent classifying spaces.

I have a piece of juvenilia on this topic, considering the case of zero-dimensional submanifolds. See Section 9 of

O. Randal-Williams, Embedded cobordism categories and spaces of submanifolds, IMRN 3 (2011) 572-608.

It concerns how close the relationship between

1. the cobordism category having oriented 0-manifolds inside $M$ as objects, and oriented 1-dimensional cobordisms in $M \times [0,t]$ as morphisms, and
2. the fundamental (topological) groupoid of McDuff's space of annihilating positive and negative particles in $M$,

is.

The main theorem in this direction is that while the categories are in no sense equivalent as topological categories (the circle as a cobordism $\emptyset \leadsto \emptyset$ is contractible as a loop in McDuff's space), they do nontheless have homotopy equivalent classifying spaces.