Let $X\rightarrow Y$ be a smooth morphism of schemes and consider the blowup of the fiber product along the diagonal, $W:=Bl\_{\Delta}X\times\_Y X$.


a.Does there exist a collection of **smooth** morphisms of schemes $X\_i\rightarrow Y\_i$ such that 


1.the schemes $W\_i:=Bl\_{\Delta\_i}X\_i\times\_{Y\_i} X\_i$ are **irreducible** and 


2.there exist morphisms $W\_i\rightarrow W$ ,such that $\{W\_i\rightarrow W\}$ is an **etale** cover?


[Here $\Delta\_i$ denotes the diagonal of $X\_i\times\_{Y\_i} X\_i$.]


b.**Same question with the additional assumption that $Y$ is smooth.**


In case $Y$ is singular, the answer in a. is negative, see Will Sawin's answer below. 



irreducible etale cover of a blowup
Let $Y$ be the nodal cubic curve and let $X$ be its finite etale double cover. Then $X \times\_Y X$ is just a union of two copies of the finite etale double cover. Blowing up changes nothing.


You have a slight problem here - the finite etale double cover is union of two $\mathbb P^1$ which intersect at $0$ and $\infty$. It has no etale cover by any irreducible schemes, because each irreducible scheme must map to only one of the two $\mathbb P^1$s, but any etale neighborhood of either $0$ or $\infty$ intersects both $\mathbb P^1$s.


So I think no $W\_i$ work, which means the answer is no.


