If sampling time is effecting mice, good luck with Human Studies. It is not easy to get human gut microbiome samples in clinical trials especially when they are already suffering from disease.
Even with longitudinal studies, patients can skip donating samples. Making it difficult to compare. Also diet (big factor) and genetics can impact your microbiome.
Having said that, I am big fan of longitudinal microbiome studies with good metadata collection such as time of sample collection, Bristol stool scale, and diet(this is very difficult to quantify).