ag.algebraic geometry – Is this property of polynomials generic?

Let $n geq 2$, and consider a polynomial $f$ in $n$ variables, say over a field $K$ of characteristic 0. Recall that $f$ is geometrically irreducible if $f$ is irreducible over the algebraic closure of $K$. We know that for $n geq 2$, being geometrically irreducible is a generic condition, i.e., applies to a non-empty open subset (in the Zariski topology) of the space of polynomials of degree $d$, say.

It seems that some geometrically irreducible polynomials are “more” reducible than others. Here is the example I have in mind: take $f(x,y) = x^3 – y^2$, so that $f$ is geometrically irreducible. However, $f(u^2,v^3) = u^6 – v^6$ IS reducible, in fact splits completely over $overline{mathbb{Q}}$.

Let us define two further classes of polynomials: we say that $f$ is practically reducible if there exist polynomials $u_1, cdots, u_n$ such that $f(u_1, cdots, u_n)$ is geometrically reducible, and we say that $f$ is algebraically practically reducible if there exist algebraic functions $u_1, cdots, u_n$ such that $f(u_1, cdots, u_n)$ is a polynomial which is geometrically reducible.

My questions are: is the condition of being “practically irreducible” and “algebraically practically irreducible” generic conditions? That is, do there exist non-empty Zariski-open subsets of polynomials of given degree which are not practically reducible/algebraically practically reducible?