It is very hard to give you a general answer - there probably is not a general case response. Many GMM toolboxes have some built in assumptions which are important to know.

For example; https://en.wikibooks.org/wiki/R_Programming/Method_of_Moments

I'm not sure how many ways there are to deal with dynamics.  

Generally, we favor the methods GEP Box called for - start out with the "mechanistic" cause and effect variables.   If you know you have some dynamic system variables, it may be best to correct for these first, and then do GMM estimation on the stable/static elements.

If you have to do dynamic regression, you may need a method designed for that, like Box Jenkins.

You have not stated the format of your panel database. Do you have small T and large N or you have large T and small N. When you have N>T, you do need to worry about stationarity of the variables because the problem of autocorrelation much less severe and can be corrected by controlling for time-specific effects (by including time dummies). So, in a system GMM framework you will go ahead and run your regressions, but you will need to control for time-specific effects. However, when T>N, then you will have to worry about stationarity and the estimation would be much more demanding. You will need to ask yourself, why do we perform unit root tests? It is because we want to establish whether the estimates are susceptible to a spurious regression. So, after unit root testing, we should perform cointergration tests to establish whether there is a LR relationship between the dependent variable and the predictors. If the cointegration hypothesis can not be rejected, then we proceed with untransformed variables. However, where cointegation is rejected, there is need to consider an error correction model. 

I think it depends on the data you are dealing with. If you use time series data the problem you raised is worrisome. However, system GMM with panel data may be not targeted to stationary problem. System GMM at first difference is good to handle autoregressive problem and if your result is not significant go further and introduce the lag of dependent variable as one of the explanatory variables to control endogeinity problem. So that you can apply Two step GMM estimation with internal instruments.

Thank you for your helpful contributions. The way I have choosen to avoid this problem is to average my data (original time period spanning from 1990 to 2014). I performed non-overlapping five years average, and I finally have five observations per country. So I think a unit root test is not yet relevant.

Eliud Moyi

Can you please share the link of the literature that supports the following:

"When you have N>T, you do need to worry about the stationarity of the variables because the problem of autocorrelation much less severe and can be corrected by controlling for time-specific effects (by including time dummies)"

I would love to read and gain more knowledge regarding this topic.

Thanking you in advance.