Aspects of the Byzantine civilization continued to spread even as the Byzantine Empire declined due to several reasons.
Firstly, the Byzantine Empire was a cultural and economic center of the Mediterranean world, and its decline did not necessarily diminish its cultural and economic influence, which has continued to spread through trade, diplomacy, and intellectual exchange between Byzantine scholars and European centers.
Secondly, the Orthodox Christianity, which was the official religion of the Byzantine Empire, played a significant role in the spreading of Byzantine cultural and intellectual legacy. The Orthodox Church established numerous churches, schools, and monasteries outside of the Byzantine Empire, which transmitted Byzantine thought, art, and literature.
Thirdly, the Byzantine Empire was also a melting pot of different cultures, and its territories were inhabited by various ethnic and linguistic groups. This diversity contributed to the exchange of ideas, customs, and practices, which continued to spread even after the decline of the Empire.
Overall, the decline of the Byzantine Empire did not mean the disappearance of Byzantine civilization and its legacy continued to influence the world in various ways.