Post-hoc tests for SEM?

Question

Are there any post-hoc tests for Structural Equation Modeling? For example, suppose you have five groups and you run a multiple-group SEM model to test for measurement invariance. If you find that only configural invariance is established, how would you test which of the five groups differ and which do not?

Answer 1

I would free everything and look at the parameter estimates. That might be clear enough.

Then I would constrain 4 groups at a time and look at the fifth. Does leaving one group out 'solve' the problem?

I guess you could test pairs of groups, but that seems like overkill and not likely to be theoretically informative.