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Not really sure what you are looking for, but the easiest might be to just add lags of your input variables in the same linear model that you are using.

If you are looking for an actual timeseries method I would checkout either darts [0] or statsforecast [1]. They are currently the most mature timeseries packages.

[0] https://unit8co.github.io/darts/ [1] https://github.com/Nixtla/statsforecast

In Machine Learning conference papers, a common approach is to model relationships between variables using Graph Neural Networks (GNNs). Using GNNs is a powerful and flexible way to go. Maybe you can give it a try!

i use GLM with my own hand-crafted features based on my knowledge of the business and the things that influence it. Works very very well

Thanks! I take it that this means a Generalised Linear Model? Could i ask for a link to a relevant article to get me started on the flavour of GLM that you recommend?