The Nice model and the Grand Tack model both show that the distance between a planet and its host star can change greatly over time. Is it possible in either of these models for the order of planets to change as well? Or is the order of planets around a star the same as when the planets formed?
Yes, the order of planets in a system can change over time. Indeed, in some variations of the Nice model, Uranus formed further out than Neptune and then switched places when Jupiter and Saturn underwent their resonant interactions. According to Tsiganis et al. 2005, one of the original three papers on the model, Uranus and Neptune switched positions in ~50% of all simulations (presumably meaning that in some where Uranus formed past Neptune, ended up further in, and in some where Neptune formed beyond Uranus, it traveled further inwards). Therefore, a number of models reproducing the parameters of the giant planets in the present-day Solar System involve Neptune forming closer to the Sun than Uranus. This seems to fit better with models of the Sun's protoplanetary disk.