The enneagram makes the MBTI look like a shining beacon of empirically based science in some ways.
That being said, there are a few (very few) studies available. One is mentioned
here. The site says:
The RHETI also compares well with the NEO PI-R test, which has become the psychometric standard for testing non-pathological personality. The NEO PI-R measures the factors of Neuroticism, Extraversion, Openness, Agreeableness, and Conscientiousness, (the "Five Factor Model"). According to Dr. Newgent, "results of testing...indicated strong correlations between the findings of the RHETI and the factors of the NEO PI-R." According to the researcher: "Results of testing for concurrent validity between the RHETI types and the NEO PI-R factors indicate that the majority of the demographic descriptive variables are invariant to the RHETI. The RHETI was able to predict the NEO PI-R factors a majority of times, with a few exceptions, regardless of the demographic variable."
Which makes me dubious that such tests actually measure enneagram type, rather than just Big Five. I could be wrong, though, not having read the details of the full study. I think that there are correlations between MBTI (and Big Five) and enneagram types, but that enneagram type is fundamentally a different thing than MBTI type or Big Five traits.
My personal opinion is that the enneagram is mostly about long term, habitually used defense mechanisms. We tend not to be aware of unhealthy defense mechanisms until they either cause major dysfunction and/or until one does therapy or otherwise gains some psychological awareness. This being the case, I think MBTI-style questions are far more likely to capture MBTI or Big Five results, rather than any deeper defense mechanisms. I think it's clear there some correlations between MBTI type (or Big Five traits) and enneagram, but I think they are fundamentally different things.
Until there is some repeatable, validated way to determine enneagram type, many kinds of studies aren't possible. I'd be very interested in studies about the validity of integration/disintegration points, but I think currently such studies are effectively impossible or impractical.
For me, I think the value of enneagram lies mostly in the description of the individual types, and the rest of it I take with a grain of salt.
I'd be happy to hear about other evidence, though.