What is the ruling on performing ablution (wudu') and prayer between the first act of relieving oneself and the subsequent discovery of traces of faeces, and is it obligatory to wait for the remaining traces to exit before performing ablution?

Istinja' is for removing the visible trace of impurity from one of the two excretory organs (the front or back private part) such that water can reach it. What water cannot reach is considered internal, and internal waste does not take the ruling of impurity until it exits. What becomes visible does take the ruling of impurity. So, if you wash the impurity that has appeared at the excretory opening, this is sufficient for you. Then you perform ablution and pray. The presence of impurity inside the abdomen does not necessitate istinja' nor does it affect the validity of ablution or prayer. Istinja' is only required for visible impurity at the excretory opening. If you perform istinja' until you are certain or think you have cleansed yourself, then you have discharged your responsibility. You are not required to inspect the area afterwards, and the default is that nothing has exited. If you are certain that something necessitating istinja' has exited, then you must perform istinja' for it. If you doubt the time of its exit, it is attributed to the last time it could possibly have exited.