The Gnostics were already in decline numerically by the third century, but events brought about the elimination of Gnostic Christianity over the centuries that followed.
Elaine Pagels says,in Beyond Belief, page 168, that Constantine ordered the return or restitution of property confiscated from the Christian church during the persecutions during the decades leading up to his reign. However, his edict only applied to "to the largest and best organised Christian group" and Constantine excluded the Gnostics. By this means, Gnostics were starved of resources and treated as outside the law. Stephan A. Hoeller says, in Gnosticism, page 156, that With the triumph of orthodox Christianity after Constantine, the Gnostic religion went underground. Hoeller goes on:
The final blow to early Gnostic Christianity came in the late fourth century, when the wave of fierce persecution burst upon the followers of the Spanish bishop Priscillian of Avila, despite the pleadings of charitable orthodox Christians. From this time on, the ceaseless hunt for Gnostics made it difficult for the Gnosis to survive in western Europe.
Richard Smoley says, in Forbidden Faith, pages 69-70, a new sect known as the Bogomils arose in the Balkans, probably as descendants of the Paulicians whom the Byzantine emperor Constantine V had resettled there in 759 CE, hoping they would convert to orthodox Christianity. The Bogomils were driven westward to Dalmatia and Bosnia, where they remained, although persecuted by the dominant Christians, until the fifteenth century. With the arrival of the Ottoman Turks, many Bogomils converted to Islam, which Smoley says "helps explain why so many Bosnians are Muslims to this day, and why the religious conflicts in the area are still so grievous." The Turks sought to ease the administrative burden by having all Christians in their newly acquired European territories subordinated to the Orthodox Patriarch, but the Bogomils had suffered centuries of persecution by the Orthodox Christians and preferred to be Muslims than subordinated to the hated Patriarch.
The Cathars of southern France and Spain had spread through Germany from eastern Europe during the twelfth century. Smoley (ibid, page 76) says "The number of Catholic priests and monks who were won over to Catharism appears to have been significant."
Pope Innocent III urged the French nobility to suppress the faith, but many nobles refused. Finally, in 1208, Innocent initiated a crusade against the Cathars. The crusade was not enough to stamp out Catharism, even with large numbers of the leaders being burnt at the stake. Finally, after total military defeat, the Cathar people were forced to demonstrate allegiance to the Catholic Church. An Inquisition was set up, to identify and, if necessary, torture and condemn people found to be practising their old faith in secret.
The historical evidence is that some Gnostics converted to Catholic or Orthodox Christianity, often out of necessity, while others accepted Islam rather than join those who had long persecuted their ancestors.