What I have noticed is sudden changes in weather gets them moving, but when its constant then they ignore it.
When its dry, and starts to rain, or when it rains for days and clears up.
The change might get them moving.
Honestly, in my opinion, wind makes a bigger difference.
When everything's moving, and its noisy, they get pretty nervous.
When they cannot smell, see, or hear, they tend to be on their feet more.
Makes them easier to see

Otherwise they just do what they normally do..
And that is bed down. (They do it anyway)
An elk eats a lot, but once it is in its first stomach, they bed down and chew their cud.
They have hollow hair, they don't care about getting wet.