Channel 4's
FACTCHECK blog has some background sources on these claims.
As you say, the real question is what exactly is it that causes the improvement's associated with the Swedish free schools.
One recent study the blog mentions, from 2008, suggested that the improvements were short term and not reflected by subsequent achievement in further higher education, i.e. University/college. So maybe exam results are improving but does that reflect a genuine improvement in education (whatever that really means)?
TTFN,
WK