​ I can recommend a few earlier posts connected to this quite large topic, which does not have one simple answer I would say.

• First off, here's a post where I'm dealing with the term of the current reconquista concept only becoming more wide-spread since 1840s in Spain. It also delves into the reconquista narrative coming up in medieval Castilian chronicles. In another post in the same thread, u/mrhumphries75 gives a great account of historical events during this period, and focusing on non-Castilian polities.

• In the first part of this answer I go into some more details on the Castilian reconquista narrative, including its justifications through the "just war" concept. After that it's about some examples of how this narrative was applied to the Americas by the Spanish.

• Here I talk about the converted Muslims (or moriscos) in Iberia after the "reconquista" period, and their eventual expulsion, in case that's interesting for you.

To put it more briefly, I'd say it's difficult to distinguish between fact or fiction regarding the reconquista, esp. due to our lack of knowledge on its origins in the earlier Visigothic kingdoms. It could be more helfpulf to see the reconquista as a historical narrative that a) goes back to medieval Castilian Christian leaders trying to legitimize their campaigns against the Muslim realms; b) because of this harked back to a Visigothic past in order to portray the campaigns as a "reconquest" despite 700 years of Muslim-Christian coexistence; c) gained more influence with Castilian ascendance; and d) influenced the expulsion of the Jewish population in 1492 and of the (converted) Muslim population up until 1611. Last but not least, elements of this reconquista narrative also proved influential in the legitimisation of Spanish colonisation in the Americas and views on their native population. Hope this helps.

Current Spain descending from the Visigothic kingdom? No serious scholar that I know claims that - at least, not in the last century or so. As /u/drylaw notes above, it was more of a historic narrative that was used in the Middle Ages to legitimise Asturian and, later, Leonese and Castilian expansion.

As to population transfers/displacement, we don't have many data on what happened prior to the forced expulsions or conversions of the late 1400s to the early 1600s. But what little evidence we do have seems to suggest these couldn't have had a lot of impact. With a couple of minor exceptions, earlier (before the late 1000s) expansion of Christian polities basically consisted of taking over abandoned wastelands and resettling them with Christian Iberians from the south or with settlers from across the Pyrenees (mostly from Gascony and Aquitaine). But once those Christian kings and counts started conquering in earnest, they had no choice but let the local Muslim population stay on favourable terms. I wrote on the origin of the practice in the last decades of the 11th century [here] (https://www.reddit.com/r/AskHistorians/comments/5idk95/were_there_muslim_minorities_in_the_christian/dbemtg7/). As you read this, bear in mind that it was the more Muslim-friendly practices that developed in Aragon that later served as a model for capitulations across the peninsula.

All in all, if I may quote this [comment] (https://www.reddit.com/r/AskHistorians/comments/69080b/was_it_common_for_muslims_and_christians_to_marry/dh2ymp2/) I wrote in another thread,

Roughly, if we were to trace descent of people living in a small town or a village in Spain or Portugal for a couple of millennia we could see that the same families, living in the same place for generations, had been Christian or Jewish under the Roman Empire and under their new Visigoth overlords, stayed Christian as 'Mozarabs' (or Jewish) after the Islamic conquest, probably converting to Islam at some point centuries later, stayed Muslim after the Christian (Re)conquest as 'Mudejars' and converted back to Christianity, becoming 'Moriscos' in the early 1500s. All this without ever moving to another town or marrying outside of their community.

Note that prior to 1492 most conversions - both from Christianity/Judaism to Islam and from Islam/Judaism to Christianity must have been more of an opportunistic adaptation rather than forced conversion.

Hope it helps - and I'll be more than happy to try and answer any follow up questions you may have.