Over what distance can weather be assumed to be constant?
Weather is subject to Tobler's First Law of Geography: Everything is related to everything else, but near things are more related than distant things.
Temperature- this changes not only by the feature, but also the day of the year. The winter months, as well as the fall and spring, will have much more drastic temperature changes. This may be due to things like fronts, but is ultimately caused by heating differences from the sun. However, a few miles can yield quite a drastic change. Things become even more complicated when elevation is involved.
Precipitation- This is even more variable. Snow can be happening on a mountain, but have rain in a valley. Even the side of a mountain can influence how much rain is seen. I have seen it rain on one side of a building, but not the other. Precipitation is a tricky thing to forecast.
Visibility There are several factors that influence visibility. Fog and precipitation are probably the most common meteorological factors. Elevation can really influence your line of sight. Obviously, visibility is also influenced by the presence of smoke as well as dust.The smaller the particle, the more likely it will obscure vision. And snow tends to be more reflective than rain. Something else that may influence visibility is wind speed, as aforementioned particles can be kicked up from the surface. So visibility will change depending on how all of these change. It may even change depending on the direction you look.

All of this to say is that it really depends and there is rule of thumb.