People can change their language. Hungary and Turkey speak the languages of conquerors who have left very little genetic trace. (France is in a similar position!) Many Ethiopians speak a Semitic language (Amharic) for reasons unknown to us. Yiddish is a Germanic language spoken by a non-Germanic people. The Chinese of Thailand often cannot speak any Chinese (they speak Thai).

I assume the argument here is that if you see a huge linguistic expansion and a huge demographic expansion occurring at the same time, you can reasonably conclude that they are the same phenomenon and the expanding language is spoken by the expanding people. But the language and the people are not in general the same; people may voluntarily adopt a foreign language without moving (compare India and English) and they may fail to bring their language with them when they move. They cannot avoid bringing their DNA with them, but it will die out when they do, which their language may not.