How is factor analysis used on time series data? Factor analysis is used for cross-sectional data where the observations are independent. How is this concept applied on time series data where the observations are not independent?

I think it really depends on the setting.
If you make a longitudinal psychological research, and have a questionnaire filled every week (Likert scale answers), then you can run a factor analysis based on the Cronbach alpha for all, then compare the results over time.
If you run sensor data, which is all numerical, it goes down to signal processing. There you can apply the same but with PCA, or apply other analyses like covariance, correlation, multivariate regression and so on.
If all your data over time is nominal, you can simply run chi-square every period and again compare the results.
It really needs context and specifics to relate to some solution.
