Ok, I understand the concept of "don't fragment". But on a GRE interface we have two options:

1) "No". Meaning "ok, fragment if needed"
2) "inherit". Meaning "inherit the value from..." From where? That's what I want to know.

Where does the value come from, if I choose "inherit"?

I'm pretty sure it means the parent device' setting if there is one, otherwise yes.

Which parent device? The GRE tunnel just exists, it isn't attached to some interface. Would it be the interface through the connection is made?

I always assumed it inherits it from the packet being encapsulated. Not sure this assumption is correct, though.

afaik, inherent indeed means "use DF flag" of original packet. Note a GRE interface is unaware of its parent interface, all you specify is an IP local/remote addresses for the tunnel

Well, it makes sense, to use the flag from the encapsulated packet.

Sorry, that was a bad wording... I meant the interface by which the connection was made through...

But I think andriys and others are right, it means to copy don't fragment from the encapsulated packet to the outer GRE packet.
(same for DSCP)