All manifolds considered here are compact and orientable. A 3-manifold (with possible boundary) is irreducible if any smooth sphere bounds a ball. Note that a closed irreducible 3-manifold is prime, and a closed prime 3-manifold is irreducible unless it's $S^1\times S^2$.
Suppose I remove a collection of thickened loops $S^1\times B^2$ from a closed 3-manifold $M$, forming a 3-manifold $Y$ with (possibly disconnected) 2-torus boundary. Or suppose I plug up such a $Y$ into a closed $M$.
Is there any relation between (ir)reducibility of $Y$ and $M$? When can I expect an irreducible (respectively, reducible) $M$ to result in an irreducible (respectively, reducible) $Y$?
I see that the irreducible $S^1\times D^2$ plugs up into the reducible $S^1\times S^2$. I also see that if I take a connected sum $M$ (reducible) and remove an $S^1\times D^2$ that cuts through the neck then, possibly, the resulting $Y$ is irreducible. I also see that if I take an irreducible $M$ and remove some thickened loops in a small ball, the resulting $Y$ is a connected sum of $M$ with a thickened link complement in $S^3$.