Ed and Keith correctly described the assumptions of the standard MIMIC model (as well as the statistically equivalent restricted factor analysis [RFA] model, in which the grouping variable merely covaries with the factor rather than predicts it). However, you can additionally test for differences in factor loadings by using a product-indicator approach to model an interaction between the latent factor and the grouping variable. Oddly enough, the covariance between the latent interaction factor and the common factor captures information about heteroskedasticity of common factor variances across groups, so it is robust to "violating" that traditional assumption of MIMIC/RFA models.
Heteroskedasticity of residual variances is not captured (unless residual covariances with product-indicators are modeled, which would render this model underidentified), but my grad student has recently found that results are quite robust to violation of that assumption as well (only minor Type I error inflation in cases of severe residual heteroskedasticity). Paper is still under review. A limitation of the product-indicator approach for latent interactions is that it is only applicable to continuous indicators ("products" of ordinal variables make no sense).
The advantage of MIMIC/RFA over MG-CFA is that single-group models are more stable in smaller samples because they estimate fewer parameters (e.g., all parameters can differ across groups in MG-CFA, except for arbitrary identification constraints, which requires more observations). However, with 5 dummy codes to represent 6 countries, you would have to calculate product indicators with each dummy code to define a latent interaction between each dummy code and the common factor(s). So that model might get so large that it loses its small-sample advantage.
If you use MG-CFA, you still get the interpretation "relative to the reference country" because the latent mean is fixed to zero in the first (reference) country. Thus, estimated latent means in other countries are merely differences from the reference country. In the semTools package, the measEq.syntax() function could be helpful for writing lavaan syntax to test invariance using MG-CFA, since you are unfamiliar with it. But I recommend reading as many textbooks, slides, or tutorials as you can find.
Terrence D. Jorgensen
Assistant Professor, Methods and Statistics
Research Institute for Child Development and Education, the University of Amsterdam