Assuming you were within an eruv would you be able to carry around a tracker? You do not turn the device on and off but it can communicate with other tracking devices as you walk.
I believe since everything the airtag/tile does is completely not discernible to the naked eye i.e. you can not see it doing anything, there is no problem on Shabbos as a result of the electronic elements. Similar to why there is no issue opening a refrigerator even though there are microprocessors inside measuring temperatures and time the door is open etc.