In general, I think this should fail for most non-smooth local complete intersections. For a specific example, let $X = \mathbb V(xy) \subset \mathbb A^2$ and $Y = *$. Then the stalk at the dualizing complex of $X$ at the origin is $\mathbb Q_l \oplus \mathbb Q_l[1]^{\oplus 2}$, contradicting the local constancy of $f^! \mathbb Q_l = \omega_X$.

In general, I think this should fail for most non-smooth local complete intersections. For a specific example, let $X = \mathbb V(xy) \subset \mathbb A^2$ and $Y = *$. Then the stalk at the dualizing complex of $X$ at the origin is $\mathbb Q_l[1] \oplus \mathbb Q_l[2]^{\oplus 2}$, contradicting the local constancy of $f^! \mathbb Q_l = \omega_X$.

I think this should fail for most non-smooth local complete intersections. For instance, let $X = \mathbb V(xy) \subset \mathbb A^2$ and $Y = *$. Then the stalk at the dualizing complex of $X$ at the origin is $\mathbb Q_l \oplus \mathbb Q_l[1]^{\oplus 2}$, contradicting the local constancy of $f^! \mathbb Q_l = \omega_X$.

In general, I think this should fail for most non-smooth local complete intersections. For a specific example, let $X = \mathbb V(xy) \subset \mathbb A^2$ and $Y = *$. Then the stalk at the dualizing complex of $X$ at the origin is $\mathbb Q_l \oplus \mathbb Q_l[1]^{\oplus 2}$, contradicting the local constancy of $f^! \mathbb Q_l = \omega_X$.