In the USA, if the mail is endorsed with "service requested" or bears first class postage, you can write the reason and the mailpiece will be returned to the sender.
However most non-profits use the non-profit postage rate or "standard mail", which includes no such return feature. Returning the mail to the carrier does no good, as it will just be discarded and more of the same will come.
The USPS recommends using the Direct Marketing Association's "Deceased Do Not Mail" list. See https://www.usps.com/manage/mail-for-deceased.htm for a link. There's also a "Do Not Contact List for Caretakers" in case you are caring for an elder or dependent. Start early, as it takes months and months to take effect. And this only is processed by members of the Direct Marketing Association.
I did find that a few key organizations do most of the list selling. Call those organizations directly, and the mail will slowly trickle off.
Data broker Acxiom has a deceased registration process. The top data brokers are as of this writing Acxiom, Corelogic, Datalogix, eBureau, ID Analytics, Intelius, PeekYou, Rapleaf, and Recorded Future.