What is the ruling on Muslims in their countries differing regarding the beginning and end of the month of Ramadan, and following Saudi Arabia in that? Is their action correct?

The preponderant view is to consider the divergence of moon-sighting horizons. Thus, the sighting of the crescent moon in one country does not obligate another country whose horizon differs. So, if a person's country's horizon differs from another country, they should not fast with them. If they do fast with them, they should continue fasting with the people of their own country, even if they end up fasting 31 days. However, if one adheres to the view that does not consider the divergence of horizons, then there is no impediment to fasting and breaking fast with people from outside their country, while keeping this discreet if its disclosure would lead to discord or strife.