I've had delayed luggage from Emirates, KLM, and Air NZ. They all offered some degree of on-the-spot compensation to tide you through. Emirates gave a tiny 68 NZD prepaid debit card (2009), KLM a bit better 100 SGD in cash (2010), and I forgot what Air NZ gave (it can't have been much and was over 10 years ago now). This was for the initial delay. Keep receipts for anything purchased during the period without the luggage (clothes, food, toiletries, phone bills, etc... pretty much everything) and claim additional costs with the airline directly afterwards.

KLM lost our luggage that had clothes for a wedding we were attending the day after the flight. I was pleasantly surprised when they paid out the receipts for having to purchase a new suit for the wedding (Not an expensive suit but still a couple hundred quid, and this was in Singapore too). They took 3 months to process the claim which was frustrating and disappointing, but they came through in the end which was good.

But as already pointed out, if the luggage is delayed en-route to your home location, I don't think the airline has to give anything. You may get some goodwill miles out of it.