Let $k$ be a field. Let $X$ be the Hilbert scheme of subschemes of $\mathbb{P}^n_k$ with a specified Hilbert polynomial. Let $Y$ be another Hilbert scheme of subschemes of $\mathbb{P}^n_k$ with a different specified Hilbert polynomial. An example I have in mind is $n=3$ and $X$ the space of lines in $\mathbb{P}^3$ and $Y$ the space of cubic hypersurfaces in $\mathbb{P}^3_k$.
Consider the incidence variety: $\{(x,y)\in X\times Y: x\subseteq y\}$. The constructions of this incidence variety I have seen have always used equations, but this seems less easy if the schemes parametrized by $X$ and $Y$ are cut out by many equations. It is not even clear to me whether this incidence "variety" should even be nonreduced in all circumstances. My question is then, is there a good equation free way to scheme theoretically describe the incidence variety?