To what extent is the hadith "The Messenger asked Gabriel, 'Will you descend after me?' He replied, 'Yes, ten times'" authentic?

We have not come across the exact wording of the inquired-about Hadith. However, if it is authentic, its meaning is that Gabriel (peace be upon him) will descend to Earth after the death of the Prophet (peace and blessings be upon him). But his descent will not involve conveying revelation to humans, because revelation ceased with the death of the Prophet (peace and blessings be upon him). He is the Seal of the Prophets, as Allah Almighty said: (Muhammad is not the father of any of your men, but he is the Messenger of Allah and the Seal of the Prophets. And Allah is ever All-Aware of everything.) [Quran 33:40]. And because the Prophet (peace and blessings be upon him) said: "And there is no prophet after me." This is a point of consensus among Muslims.