Here's the method I had in mind looking at the examples I had. Let's Let $X \subset Y$ be a smooth hyperplane section. Our goal is to detect cycles on $X$ that are not complete intersections of $X$ with cycles on $Y$.
Let's consider a cycle $Z$ on $Y$ such that $Z \cap X$ is reducible, say $$Z \cap X = A + B$$ for some cycles $A$, $B$ on $X$. Then we may hope $A - B$ or some similar combination can be a vanishing.
For quadrics and cubics above we take $Z$ to be tangent linear subspace of appropriate dimension.
Has anybody seen something like that applied in other cases?
The sad thing is that I sort of can make it work for $MG(3,6)$ to describe a vanishing cycle, but it's unclear from the description I get whether the cycle is rational or not.