A detailed discussion of dynamic regression models is provided in Pankratz (1991).
A generalisation of dynamic regression models, known as “transfer function models,” is discussed in Box et al. (2015).
Bibliography
Box, G. E. P., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. (2015). Time series analysis: Forecasting and control (5th ed). John Wiley & Sons. [Amazon]
Pankratz, A. E. (1991). Forecasting with dynamic regression models. John Wiley & Sons. [Amazon]