My goal is to analyse how some variables play a part in clustering a group of individuals, i.e. what are the deterministic factors that differentiate one cluster from another. Besides manually analysing the distribution of variables in each cluster, I was thinking about using classification techniques like random forest on the dataset, using the clusters as a pseudo response, which might give some idea of variable importance. However I am not sure if this is a sensible thing to do, and there seem to be no such discussion anywhere.
I think any method to study variable importance is worth a try. This could be as simple as the correlation between features and binary indicator variables for each cluster, or as complex as a random forest.
Often people don't do such an analysis, and then won't realize that (e.g., because of inappropriate preprocessing) their clusters only depend on a single variable. So in particular, you should also check which variables do not have an impact.