So, (nearly) a century old ghost of 'Pirenne thesis' still lingers in the 7th century East Mediterranean.

The recent trend of this topic, i.e. the continuity and the break of the 'Roman' exchange networks during 'Late Antiquity' into the 7th century in the East Mediterranean, pay attention to the following point: To what extent the private trading sector, i.e. commerce, especially of luxury, played an important role in the total amount of exchanged goods as well as the whole 'economic' activity at that time?

Wickham points out the difference between the Byzantine and the Islamic (Umayyad) taxation system and its possible impacts on the break down in the 7th Eastern Mediterranean (Wickham 2005: 713-18, 759-94).

The fiscal system of Eastern Roman-Byzantine Empire, such as annona grain export, had been somewhat 'centralized' at Constantinople as its center, though the degree of their dependency on this kind of fiscal transfer within the total exchanges was still less than the 5th century Western Mediterranean where the establishment of the Vandal kingdom in the North Africa seriously disrupted the annona system itself. Once Byzantine Empire lost the political control of provinces of Syria, Palestine, and Egypt in course of the 7th century, the fiscal ties between the capital and these provinces died out, only the commerce remains.

Wickham summarizes the possible situation as following:

'The substantial weakening of exchange links between the Levant coast and Egypt in the late seventh century is one instance. Although these regions were both part of the caliphate, they had no fiscal links, and the basic underpinning of large-scale exchange had gone; only a small number of Egyptian exports continued to go to Caesarea. The end of the fiscally supported world-system, as we have seen, spelt local crisis or stagnation for the sectors of the coast most dependent on it, the Antioch and Gaza areas, which match the Spanish coast and most of Tunisia in the degree of their dislocation. Conversely, the revival of interregional networks of in the 9th century maps onto 'Abbadid fiscal recentralization (Wickham 2005: 779).

Thus, even the exchanges between the conquered regions in the Eastern Mediterranean under the same political authority of the caliphate could be stagnated due to the lack of the centralized fiscal system. The rise of Abbasid in the middle of the 8th century could also drastically reversed this socio-econoic trend again.

On the other hand, McCormick suggests the possible driving force of the switch of dirham flow channel into Europe from the Mediterranean to the Russian waterways.

• The Europeans, especially Carolingian Franks sold their POW as slaves to the Middle East (McCormick 2001: 733-77), though I suppose the end of the political expansion of the Carolingian Franks (ca. 800) probably limited the supply of new slaves.
• Prior to the first local (Khazar) silver coin in the 830s, Abbasid dirhams already flowed into the Russian steppes and waterways like Dons, represented by the hoards there, probably in exchange for slaves and furs. Ibn Khurdadhbeh (middle of the 9th century) was the first written testimony of this northern trade route.

References:

• Franklin, Simon & Jonathan Shepard. The Emergence of Rus 750-1200. Longman: London, 1996.
• Kovalev, R.K. and Kaelin, A.C. 'Circulation of Arab Silver in Medieval Afro‐Eurasia: Preliminary Observations'. History Compass 5 (2007): 560-580. https://doi.org/10.1111/j.1478-0542.2006.00376.x
• McCormick, Michael. Origins of the European Economy: Communication and Commerce AD 300-900. Cambridge: Cambridge UP, 2001.
• Pirenne, Henri. Mohammed and Charlemagne. (1937).
• Wickham, Chris. Framing the Early Middle Ages: Europe and the Mediterranean, 400-800. Oxford: OUP, 2005.

The case of Caesarea Maritima is not very indicative. In many ways, its actual story can be similar to that of the poor library of Alexandria: Some believe it was burned by Muslims, but it more likely declined for centuries beforehand. For Caesarea Maritima, the Muslim conquest was certainly a shock, but Kenneth G. Holum convincingly argues in a recent article that "the flight of the Christian aristocracy" from the conquerers and "more gradual economic change" is a better explanation for the downfall of the port.

Regardless, the rise of the Muslims led to the disruption of some trade routes but to the flourishing of many others. Unlike early Christianity, Islam did not denounce trade but supported it. It must have: the Quraysh tribe from which Muhammad hailed was a tribe of merchants that monopolized routes linking Mecca and Syria, like many other tribes that could not otherwise finance their own subsistence. One of the concluding chapters (Surahs) of the Quran, named after the tribe of Quraysh, speaks about He who makes "Quraysh habitually secure -- secure in their trading caravan to Yemen in the winter and Syria in the summer".

The clearest evidence of the major role the Arabs and their Islamic empire (which integrated previous routes and forged new ones) had on east-west trade was the proliferation of their coins throughout the region, coins that are now dug by archaeologists all over the place. As Alexis C. Kaelin and Roman K. Kovalev show in their discussion, 1656(!) hoards containing almost 500,000 dirhams were "deposited[...] in northern Europe from c. 800 to c. 1100[...] mostly through Russia". But they were not big sailors at first, even though they came to dominate much of the Indian Ocean trade with the Mediterranean world later on, which meant that the Age of Discoveries was particularly painful for their successors, the Ottomans.