TL;DR: They do indeed! The details vary, but certainly after Geoffrey of Monmouth, it's generally agreed that Mordred was responsible for Arthur's downfall - Arthur killed Mordred, but was mortally wounded in the process.

c. 1136: The original account in Geoffrey of Monmouth's Historia Regum Brittaniae, the generally accepted first appearance of Arthur, in which he refers to Mordred as Modred, says:

[In Cornwall] Modred, as he was the boldest of men, and always the quickest at making an attack, immediately placed his troops in order, resolving either to conquer or to die, rather than continue his flight any longer. [...] It would be both grievous and tedious to relate the slaughter, the cruel havoc, and the excess of fury that was to be seen on both sides. In this manner they spent a good part of the day, till Arthur at last made a push with his company, consisting of six thousand six hundred and sixty-six men, against that in which he knew Modred was; and having opened a way with their swords, they pierced quite through it, and made a grievous slaughter. For in this assault fell the wicked traitor himself, and many thousands with him. [...] And even the renowned king Arthur himself was mortally wounded; and being carried thence to the isle of Avallon to be cured of his wounds, he gave up the crown of Britain to his kinsman Constantine, the son of Cador, duke of Cornwall, in the five hundred and forty-second year of our Lord's incarnation.

c. 1155: Wace's Roman de Brut (translation by Eugene Mason) says:

Arthur was sick with wrath that he was not avenged of Mordred. He had neither peace nor rest whilst the traitor abode in his land. Arthur learned of Mordred’s strength in Cornwall, and this was grievous to him. [...] Arthur sent after his men to the very Humber. He gathered to himself so mighty a host that it was as the sand for multitude. With this he sought Mordred where he knew he could be found. He purposed to slay and make an end of the traitor and his perjury alike. Mordred had no desire to shrink from battle. He preferred to stake all on the cast, yea, though the throw meant death—rather than be harried from place to place. The battle was arrayed on the Camel, over against the entrance to Cornwall. [...] There, too, was Mordred slain in the press, together with the greater part of his folk; and in the selfsame day were destroyed the flower of Arthur’s host, the best and hardiest of his men. So the chronicle speaks sooth, Arthur himself was wounded in his body to the death. He caused him to be borne to Avalon for the searching of his hurts. [...] Arthur bade that he should be carried to Avalon in this hope in the year 642 of the Incarnation. The sorer sorrow that he was a childless man. To Constantine, Cador’s son, Earl of Cornwall, and his near kin, Arthur committed the realm, commanding him to hold it as king until he returned to his own. The earl took the land to his keeping. He held it as bidden, but nevertheless Arthur came never again.

c. 1400: In the Alliterative Morte Arthure (because who needs prose when you can have it in stanzas?):

He hit Mordred amid the breste And out at the backe bone him bore; There hath Mordred his life lost, That speche spake he never more; Then keenly up his arm he cast And gave Arthur a wounde sore, Into the hede through the helm and crest, That three times he swooned there.

Wider context in Alliterative Morte Arthure: I've found references stating that Mordred attached Arthur in disguise in this version, but that Arthur recognised him because he was using Arthur's other sword (not Excalibur), Clarent. However, I need to read the orignal text more carefully to verify this.

1485: Thomas Mallory, in his famous Le Morte d'Arthur (Helen Cooper, Oxford World Classics version), in Book VIII: The Death of Arthur, says:

And when Sir Mordred saw King Arthur, he ran unto him with his sword drawn in his hand; and there King Arthur smote Sir Mordred under the shield with a foin of his spear, throughout the body, more than a fathom. And when Sir Mordred felt that he had his death’s wound, he thrust himself with the might that he had up to the bur* of King Arthur’s spear; and right so he smote his father, King Arthur, with his sword holding in both his hands, upon the side of the head, that the sword pierced the helmet and the tay* of the brain. And there-with Mordred dashed down stark dead to the earth. And noble King Arthur fell in a swoon to the earth, and there he swooned oftentimes.[...] ‘Comfort thyself,’ said the King, ‘and do as well as thou mayest, for in me is no trust for to trust in. For I will into the vale of Avilion to heal me of my grievous wound; and if thou hear never more of me, pray for my soul.’

Wider context in Le Morte d'Arthur: Sir Gawain has been slain in a recent battle with Mordred's forces. Later, he comes to Arthur in a dream and warns him not to battle with Mordred. Arthur asks for a truce, and the two armies meet to discuss this, but after a snake appears, a knight draws a sword to kill it. This is taken as a symbol of aggression, and the "final battle" starts. After Arthur is mortally wounded, he gives his sword, Excalibur, to Sir Bedivere to throw in a lake. After Sir Bedivere finally complies, a hand rises from the water to catch the sword.