Also called accretion . The neutron star has a disk of matter near it that is falling on to it - accretion disk. These X-ray binaries have huge phenomenology. So far we have talked about 2 kinds of dead stars - white dwarfs and neutron stars. 3rd type is even more extreme - can summarize in table.
Nothing can hold the star up.
Shrinks to size of a speck of dust and even smaller. As far as we know collapse continues until we can't use the current laws of physics to describe it any more - singularity. Events going on inside this radius cannot affect our Universe since they can't escape. Not cosmic vacuum cleaners - don't have a magic ability to suck in matter only gravity. Far away the gravitational force is essentially the same as that for a star of the same mass.