For topological reasons you can see that any minimal surface $\Sigma\subset \mathbb{S}^3$ that is not a sphere or a torus has to give rise to a mean convex flow that becomes singular before it disappears. This is essentially because the only possible singularities are spheres and cylinders which prevents the flow from collapsing all at once when the genus is larger than one. It's possible that to make this fully rigorous one needs to a result of Colding-Minicozzi. Who show that in this setting the singular set has to be a closed C^1 curve (note there can't be any spherical singularities as that would imply a disconnecting singularity at an earlier time).

It's not hard to generalize this to higher dimensions.

It should also be possible to show using that if there are small enough necks then the perturbed flows have to have neck pinches by more localized arguments (at least in the inward direction). If you widen the necks then it is a bit trickier, but there is still a neck pinch the other way as the surface widens.Think of a torus, the inward perturbation has a punch of neck pinches, but the outward perturbation has a neckpinch that turns things into a sphere (its helpful to think of the torus as the boundary of a sphere with a small hole drilled into).

For topological reasons you can see that any minimal surface $\Sigma\subset \mathbb{S}^3$ that is not a sphere or a torus has to give rise to a mean convex flow that becomes singular before it disappears. This is essentially because the only possible singularities are spheres and cylinders which prevents the flow from collapsing all at once when the genus is larger than one. It's possible that to make this fully rigorous one needs to a result of Colding-Minicozzi. Who show that in this setting the singular set has to be a closed C^1 curve (note there can't be any spherical singularities as that would imply a disconnecting singularity at an earlier time).

It may be hard to generalize this to higher dimensions.

It should also be possible to show using that if there are small enough necks then the perturbed flows have to have neck pinches by more localized arguments (at least in the inward direction). If you widen the necks then it is a bit trickier, but there is still a neck pinch the other way as the surface widens.Think of a torus, the inward perturbation has a punch of neck pinches, but the outward perturbation has a neckpinch that turns things into a sphere (its helpful to think of the torus as the boundary of a sphere with a small hole drilled into).