We do not recommend that you use these weights. Instead use the maximum likelihood ability to deal with the missing values. If all observations are present at wave 1 and sampling weight is available for that wave we recommend that you use that weight variable. Weights for the later waves should be ignored. They are designed for univariate analysis and not for growth modeling.
The selection probability for a longitudinal model is done at the time of selection (wave 1). Wave 2 etc weights typically are computed to account for missing data and are no longer the inverse of the selection probabilities. We recommend that you use FIML based principles to account for the missing data rather than weights.
I understand what the weights are for, my question is whether it is possible to specify a design weight in a GMM. To simplify things, let's assume there is 100% response at each wave but that the sample was drawn such that there are 2 strata and the sampling fraction within each stratum was disproportionate, i.e. units within stratum 1 are over-sampled. Now, if we want to make inference to the target population, we must apply a design weight to correct for the disproportionate stratification. Is it possible to do this?