Randomly Assigned First Differences?

We consider a first-difference regression of an outcome evolution $\Delta Y$ on a treatment evolution $\Delta D$. If the treatment effect changes over time, the regression residual is a function of the period-one treatment $D_{1}$. Then, researchers should test if $\Delta D$ and $D_{1}$ are correlated: if they are, the regression may suffer from an omitted variable bias. To solve it, researchers may control for $E(\Delta D|D_{1})$. We revisit Acemoglu et al (2016), who study the effect of imports from China on US employment. $\Delta D$ and $D_{1}$ are correlated. $\Delta D$'s coefficient is less negative when controlling for $E(\Delta D|D_{1})$.