They wouldn’t be my choice of words but she needs to be able to speak freely. We talk about home invasions when those an undertaken by two or three people so that doesn’t imply a great number of people. Influx would be an equivalent word. I don’t see it as dehumanising.
Swarm is probably less useful a strict description of events but again I don’t consider it a particularly loaded word. It does invoke the sense of a large number that overwhelms attempts to repel them. Isn’t that a fair description of what is happening - that a significant number are arriving and we can’t deal with them effectively? The backlog of asylum applications is evidence of that. I don’t think that reflects on the individual migrants, it is more a comment about the state of the system.
Getting back to the issue in hand, I don’t think the Nazis used these words (or their direct equivalents in German) to describe a situation where people were attempting to enter Germany from the outside so that undermines Lineker’s point. The situations simply are not equivalent. The Nazis are not considered one of the most evil regimes in history because they had a robust approach to asylum seekers. It is a false, and therefore offensive, equivalence.