It is very normal for data scientists and modeling professionals to be concerned with the stability of the model. It basically means that if a variable is important today, it cannot change its importance over time because this instability would be a demonstration of the model's weakness.
It is difficult to argue that stability is good. But in the area of credit modeling (which produces credit scores, for example) people are cracked by stability, to the point of preferring models with less discrimination to a more stable one.
My question is what is the true utility of stability and what is the correct way to make this trade-off with the discriminatory metric of the model (KS, Gini, AUC, etc.).
I would guess (personal guess) that stability somehow ties the model to a less than ideal metric because it does not allow behaviors to change over time. If they don't really change, there will be no problems, but if the effects of the variables are changing over time, it is normal to assume that this is reflected in less stability and analysts work around this by placing simpler models or removing the variables.