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If I initially did not claim my luggage after arriving from a flight, what steps do I need to take to get my luggage back?

Ideally an answer would cover all airports and / or airlines but if that turns out to be specific to either then this will be accepted as an answer in itself. I am asking out of general interest but assume such a situation could frequently arise in e.g. one of the following contexts:

• 
  Short-checking part of one's luggage and not having time to claim it due to a short connection.


• Being able to pass immigration only on the next day.


• Having any other type of impasse that does not allow waiting for the luggage to arrive.

Bonus question: I would be especially interested if it is possible to mandate someone else to receive said luggage and if so how this is done?

It depends on who is at fault for you not being able to claim.

If it is the airline's fault, so they didn't load the bags / bags got lost / bags didn't make a connection etc, you must speak to the airline staff or their ground handler before leaving the baggage hall. They will get you to fill out a form about your bag (size, colour, make, identifying features etc, plus your details). The form will include (in almost all countries) a customs waiver/release. When your bag is eventually found / turns up, the airline will take the bag through customs (assuming customs don't find something in the bag of interest...), then get the bag to you.

However, if it's not the airlines fault, or you left the baggage hall without completing the missing baggage form, or if customs have taken an interest in your delayed bag, you first need to speak to the airline staff at the airport or their ground handler. (Note - likely not airline customer services, they probably won't know, usually needs to be people actually at the airport in question). They will let you know where to go, and what times you can arrive (it may only be possible at some times of the day)

Once you're at the airport, you then contact the baggage staff (maybe via the airport information desk, maybe by phoning them, maybe by using a special phone, maybe by going to a landside desk and having the phone someone, it varies - that's why you need to speak to someone before getting to the airport!). You'll then be escorted to airside, often through a staff security channel. You will almost certainly need photo ID, often a passport, and your details will likely be recorded in a log somewhere.

Once airside in the baggage hall, you'll head to the baggage queries desk for your airline or their ground handler, the place you'd go to report missing bags. They will likely want your baggage receipt tags (the bits stuck to your passport or boarding pass at checkin). They will then go and get your bag, often from a locked storage area somewhere airside. You'll sign some paperwork, and if it's your fault possibly pay a storage fee/fine. You and your bag then exit through customs, who quite possibly will pay more attention to you and your bag than they normally would for a typical traveller, so allow extra time!

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The easiest way to avoid all of this faff is to ensure you either collect all your bags on arrival (if they're there), or speak with the airline / handling agents before leaving the baggage hall to properly report any missing/delayed luggage + fill out the appropriate forms then!