A Question About the Election of the Apostle Mathias

A reader writes:
In Acts 1, the selection of Mathias as the replacement Apostle is discussed. It talks about casting lots, and it falling on him. Do we know more about what that means and is there evidence of The Church using a similar process to make decisions through the centuries?

This is, as far as I know, the only moment in the Church’s history where the Magisterium makes a serious judgment call by this means. It’s a method that harks back to the Urim and Thummim used by the priest of the Old Covenant. By Acts 15, the Church has already settled into the habit of calling a council or synod in union with the Pope (Peter) and/or the local bishop (James) to iron out questions of navigating the signs of the times.