Dilation and erosion, in general. In mathematical morphology, growing an object is called dilation and shrinking it erosion. Take a small disk, the structuring elementLoading..., and roll it along the outside of the object's edge: everything it sweeps joins the object. Rolled along the inside, it carves the object down. The Engines Reference doesn't say how StrataQuest measures the radius, so this is the textbook picture, and the morphological operationsLoading... page covers the family.
What growing can't do. Erosion and dilation aren't inverses. Shrink a small object until it disappears and no amount of growing brings it back, which is what Shrink (retain last pixel) guards against. Growing also can't restore boundary detail lost at detection, so choose radii from the biology you want to capture, not to patch a detection.
When neighbors collide. Grow two nearby nuclei and their grown masks can reach the same pixel. In image processing generally, one common rule gives each contested pixel to the nearest object, which divides the space between nuclei along the midline, much like a Voronoi diagram. StrataQuest's Basic Operations Module documents that rule for its Grow labels operation, where each pixel is assimilated by the closest label. For Grow (global) the Engines Reference doesn't say how a collision is resolved.
Why rings help. A cytoplasmic or membrane marker contributes little inside the nucleus, so its mean over the nuclear area mostly measures whatever happens to overlap it. A ring that starts at the nuclear contour samples the zone around the nucleus instead. Without a membrane stain, the ring's width is a guess about how far the cytoplasm reaches. The MembraneLoading... engine, and the identified mask of the Cellular engine, let a stain set that boundary instead.
Growing an object is like rolling a coin around its outside edge and adding everything the coin sweeps; shrinking rolls it around the inside. Shrink a small object too far and it's gone for good, which is why one mode keeps a last pixel. A ring around a nucleus measures the zone where cytoplasmic and membrane markers sit.