Envelope Theorem in Hopkins and Kornienko (2010)
From the first order condition for optimality, we know that (equation (4) in the paper):

$$U_x(x,Z(r)-x,S(r))x'(r) - U_y(x,Z(r)-x,S(r))x'(r) + U_s(x,Z(r)-x,S(r))S'(r)=0.$$ 

When differentiating ${\bf U}(r)$ with respect to $r$, we find your expression:
$$U_x(x(r),Z(r)-x(r),S(r))x'(r) + U_y(x(r),Z(r)-x(r),S(r))(Z'(r)-x'(r)) + U_s(x(r),Z(r)-x(r),S(r))S'(r).$$ 

Replacing the first equation into the last one yields
$${\bf U}'(r)=U_y(x(r),Z(r)-x(r),S(r))Z'(r).$$ 

Now, $Z'(r)=1/g(Z(r))$ which gives equation (8).