moving solid-state drive to new macbook
You can't "transfer the drive" as you describe from a 2013/14 MBA to a "new" (assuming 2018) MBA because the SSDs on the new unit are soldered onto the logic board.

To transfer your data, you have the following options available, ideally, using Migration Assistant


connect your new MBA to old MBA over the network
utilizing a Time Machine Backup
put your 2013/14 MBA SSD into a USB enclosure


Can these things be done at an Apple store?  The first two definitely can.  It's highly unlikely they'll stock a USB enclosure.  Also, keep in mind that since you can't upgrade the SSD on the new MBA, you must configure it with sufficient capacity (minimum 500GB) to fully accommodate your needs.