It’s not so much a convention as an explicit model that says it’s constant. If you use the model
\dot{x}_d = 0 + w
for your low-frequency disturbance, i.e., an integrator of noise, you are explicitly modeling something that has zero deterministic dynamics (constant), only random fluctuations that are equally likely to go in any direction. There are more elaborate disturbance models that do not have zero dynamics, and in that case, simulating those forward in time (for forecasting) does not in general lead to a constant evolution.
diverge in what sense?
Whenever you have unmeasured disturbances you are typically not interested in pure simulation performance, there must be some mechanism present to account for the unmeasured disturbances, and the only information available about this is the measurements you do have access to. These must feed into the dynamics somehow, which is what the filtering accomplishes. The approach from your previous post on the subject sounds spot on what you want, filtering with 24h forecast at each point, minimizing the forecasting errors along the trajectory.
You can treat forecasted values (from e.g. a weather forecast) as “measurements” and incorporate those as any other measurement. Typically, those would have a rather large covariance compared to actual sensor measurements.
So am I, at least in the sense that I studied EE 