Let $f:X\to $ Spec $\mathbb Z[1/n]$ be a morphism of finite type. As $f$ is generically flat, there is a finite set of primes $S$ such that $X\to $ Spec $\mathbb Z$ is flat over Spec $\mathbb Z - S$. (This is easy to show, so let's assume $f$ is flat now.)
If $f$ is generically smooth, then there is a finite set of primes $S$ such that $X\to $ Spec $\mathbb Z[1/n]$ is smooth over Spec $\mathbb Z - S$.
You can't weaken finite type to locally of finite type here.
To prove this, you can proceed in many ways. One way is to use the sheaf of differentials. That sheaf is locally free if and only if the morphism is smooth.