The prefix el- has never really been "away". Rather it's a prefix that indicates that a motion is done to completion. Which can be on occasion expressed with "away" in English, if the starting point of the moving is given.
- Elmegyek a háztól. - I go away from the house.
- Elmegyek a házhoz. - I go to the house.
In this sentence you have "az épületből" - out of the building. It wouldn't sound too nice with "away" put somewhere in there, so I'd just say "The bank is moving out of the building."
The given English translation is slightly off. Yes, you could put "away" neatly in there, but "from the building" sounds more like épülettől than -ből.
It's still moving, so the perfect tense doesn't fit here. Aspects are a little difficult if you haven't grown up with them but I'll try to explain with a different sentence.
- Látogatom a lányt. - I visit the girl. Here the verb has no prefix, so it indicates it's an ongoing process. I have been visiting her, and I will visit her time and again. Maybe I'm not even currently at her place, but I'll surely go sometime again.
- Meglátogatom a lányt. - I am visiting the girl. Here we have a prefix, so it indicates that this is a one-time thing. Or I'm focusing on this one time I'm visiting her because this one time is special.
Similarly, elköltözik here tells us that this is a one-time thing that's going to be completed in the future. The bank moves once and then they're done. They will have left the building.
And again: "az épületből" means "out of the building". Not "from the building". So taken together I recommend a translation of "The bank is moving out of the building."
If you want to see something with the infinitive (ongoing, incompleted) aspect of költözik, I can give you "Városból városba költöztem" - "I moved from city to city". It was a long, repeated process.