# Regressing across multiple different time series using exogenous variables?

To make this situation clear, I'll use a somewhat silly, but conceptually simple example. Imagine I record teams of movers carrying furniture down the block. I measure the furniture's position/speed over time (say every 15 seconds), the size of the furniture (small, medium, or large), and the numbers of people carrying the object at each 15 second time point. People on the teams join in and help transport or let go and step back freely; maybe they join when they feel they could be useful or something, who knows, it just changes.

I record these trips multiple times across different furniture sizes, etc. I now want to determine HOW the number of movers carrying the furniture and the size of furniture affect the speed of transport. If there were only one trip, I think I would use some sort of ARIMAX model and regress the number of movers against their instantaneous speed over the single time-series run (with ARIMA errors). However, I want to make a generalization ACROSS all these recordings and I want to see if the categorical variable of furniture size has an effect as well (or how it interacts with mover number). How in the world do I incorporate all of these recorded trips (many different time-series iterations) in one analysis?

Additionally, imagine I think both the speed and the number of helpers are non-stationary, but in different ways. I think it's likely that the speed increases throughout the trip, and the number of workers increases initially (as they try to get it going, say) and then decreases (once it's moving along, maybe extra movers just get in the way). What would I do in that situation?

Lastly, I'm trying to do this in R. Crazy bonus points if you can explain it using R code!