The purpose of pseudo-out-of-sample testing is to replicate real-life conditions in a controlled experiment. So, once you are done testing your models, what will you do when you actually have to produce real forecasts? Will you select a new model each time a new observation comes along? Or choose it once and then never touch it again? Or maybe at fixed intervals, or when a forecast comes out that "looks bad"?
Your pseudo-out-of-sample testing should replicate exactly the strategy that you will actually use once it's time to forecast for real. If you're going to change the model at every new observation in reality, then the errors you get in testing with a fixed model won't be representative, and vice-versa.
The question of which refitting/model selection frequency is best has no general answer and depends on your data. Refitting too often could lead to overfitting, and not often enough could miss real changes in the process. You would also have to consider the time it takes to select a model and the costs involved each time.