:::"Does the source say that the children in Treviño study Basque??" Yes it does. You even go on to answer that question yourself... 

 :::"Los niños estudian en euskera.... there are ikastolas in Treviño." So there you go. The kids there study in Basque. 

 :::"Now, it is clear from other sources that a fair number of residents of Treviño were born in Álava or elsewhere in the Basque country. They surely send their children to the ikastolas." That's pure guesswork on your part. See WP:SYNTH: "Do not combine material from multiple sources to reach or imply a conclusion not explicitly stated by any of the sources." 

 :::"And how many purely local people do the same?" Why are the people who live there not local people? Why are their kids who were born there not local? If people who live there study in Basque then that's a clear linguistic link, regardless of whether they meet your definition of "local" or not. 

 :::"From the source, we don't know, perhaps none." Again, pure guesswork. 

 :::"And do you think (I'm addressing Valenciano) that that can be summarized as a linguistic link?" I don't need to make that call. The source has already made it. The source summarises it as a linguistic link.