Time-series decomposition to a base level and an effect of another feature
You are describing multivariate time series analysis, modeling the interactions and comovements among a group of time series variables.

You can start with a vector autoregression (VAR) model, one of the most successful, flexible, and easy to use models for
the analysis of multivariate time series. Then work through the decomposition of time series.

You'll have to decide if you want assume if all components time series are stationary and ergodic. Each component will have autocovariances and autocorrelations but there are also cross lead-lag covariances and correlations between all possible pairs of components.