Not really, free movement has lead to more tax take to be spent on the health care system. Health spending in every country has increased per capita (inflation

Not really, free movement has lead to more tax take to be spent on the health care system. Health spending in every country has increased per capita (inflation adjusted), in the UK it has too. But in the UK, for 40 years, been way under that of countries like France, Germany, Switzerland, Denmark, Norway, and of course the U.S. The rate of increase hasn't increased either. This isn't an issue of (typically working age) immigration - indeed that offsets the problem. It's an issue of an increasingly older population costing more per head. In the UK there's also an issue that people stay in hospital too long because of a failure in social care.