What is the ruling on women's discharges, and is it a condition to perform istinja' with water and change clothes when they occur? What is the extent of their effect on the validity of prayer and ablution, especially when performing Umrah?

The summary of what we lean towards is that these secretions are pure, but they nullify ablution.

You are not required to perform istinja' (washing after relieving oneself) from them, nor to purify whatever part of your clothes they might have touched.

If you find a time slot that is sufficient for prayer with a valid purification, then you must wait for that time when the discharge of these secretions stops. You then perform ablution and pray during that time. If they are discharged from you during prayer, you should interrupt it, repeat ablution and prayer, and similarly during circumambulation (tawaf).

However, if the discharge of these secretions is continuous, such that you cannot find a time for prayer with a valid purification, then your ruling is that of someone afflicted with سلس (sals, i.e., continuous discharge). You should perform ablution for each prayer after its time has begun, and you are permitted by this ablution to do what is only permissible with purification, as long as the time for that prayer has not expired.