The synthetic control (cohort) method is a very promising approach to causal inference that has been used in a number of interesting studies. It's particularly useful in situations where data are only available in aggregate or there is only one treatment observed.
In Abadie et al (2015), the authors claim that unobserved, time-varying confounders are essentially control for given an argument based on intuition:
"The intuition of this result is straightforward: Only units that are alike in both observed and unobserved determinants of the outcome variable as well as in the effect of those determinants on the outcome variable should produce similar trajectories of the outcome variable over extended periods of time."
I buy this argument but I'd like to see a firmer exposition of why this is the case. If one selects on pre-treatment trends in outcomes and observed factors relevant in driving outcomes, how does the synthetic control method control for both unobserved AND time-varying variables that may also influence outcomes?