Firstly, this question is working on the assumption that there were significant Carthaginian populations in their Mediterranean territories (i.e. some of modern Spain and various islands in the Mediterranean) while they were ruled by Carthage - if this is incorrect then please do say so. If this is the case as a follow up question it'd be great if you could expand on how great Carthaginian influence, at least, was in their colonies, in terms of culture, language etc., if you'd like to.
Anyway, I'm wondering what happened to these if they did indeed exist - were they expelled/killed after their conquest by Rome? Were there still traces of at least Carthaginian pockets afterwards? If so, how long do hints of Carthaginian presence or at least Carthaginians' descendants remain, such as the Punic language, Punic names, the Punic religion, or Punic styles of architecture/art/pottery/jewellery etc.? Thanks.
p.s. if you would object to the idea of a distinct 'Carthaginian' culture, just read 'Phoenician'.
There was certainly some lasting influence of the Carthaginians in Spain. As one example, in the early Empire Strabo writes about Phoenician-influenced Melqart-Heracles cults in Spain. He also discusses the Phoenicians’ role in the name ‘The Pillars of Heracles.’ (Geography 5.5-6) This is one small incident of Phoenician cultural influence surviving several hundred years of Roman hegemony. There might be other examples.
More recently, a DNA study led by Chris Tyler-Smith found low frequencies of a genetic marker linked to the Phoenicians in Spain and Portugal. (Identifying Genetic Traces of Historical Expansions: Phoenician Footprints in the Mediterranean, 2008) Much higher frequencies were found in places like Malta, Sicily and Crete, but these results suggest that some remnant of Phoenician settlers remained in the Iberian peninsula and their progeny still exist today.
Edit: changed 'Phoenician culture' to 'Phoenician cultural influence'
May I ask a follow-up?
Since both were launched from similar regions of North Africa, was the Moorish conquest of Spain in any way motivated by memories of the Punic wars and lost colonies in Iberia?