"I live at sea level so if i setup the kestrel there completely and i go on a hunt 400km away, will the kestrel automatically take my new enviromentals and adjust the solution to the new location?"
Answer: yes and no; you specified a location 400km away, but not its elevation (new place could also be sea level at same latitude).
Yes, the Kestel AB can store your presets for your bullet weight, BC, height above bore, etc. for use at your new location, and would be pretty close if your elevation was similar and not deviated too much from your current latitude.
No, you will not have exactly the same info auto update (its not a GPS, but does have azimuth compass capability), and ideally, you measure your MV at your new location. As you mentioned, you were making all these considerations for hunting. Most ethical shot ranges would not be too adversely affected by only taking environmentals at your new location without doing anything else.
"Will it make adjustments to my solution on its own after capturing new enviromentals?"
Again, yes and no. No, because it does not have GPS geolocation capability (you have to enter that data) and does not "know" where you are relative to your last location. Yes, because it will shift your data output based on the new local environmental data even though you did not change other settings. Your result would be good enough for 'government' work, so to speak.
For ELR, people have different definitions on just what that is. Cal Zant of the Precision Rifle Blog shows most (
https://precisionrifleblog.com/2020/08/27/best-rifle-elr-caliber-cartridge/) people fall into two camps: (1) under and (2) over 1500 yards with the latter being considered as the start of ELR ranges. Having taken some courses with Frank Galli and Marc Taylor (SH owner/Marine Scout Sniper and his Marine Scout Sniper Instructor), their recommendation was that focus on the Kestrel's small specifics (aerodynamic jump, spin drift) was negligible. That on a practical note (even if statistically significant) at non-ELR distances, these considerations most definitely were not for typical hunting ranges that provide ethical amounts of energy on the game target. The concept of ELR hunting is typically an oxymoron and