A Catholic Land

Public/Private revelation.

How does the Church deal with apparitions? What processes do the investigations follow?

The approach of the Church to investigating apparitions sees Her follow a rigid process of investigation and discernment, with the local Bishop overseeing all investigations and making the definitive ruling. Various levels of further, fuller, hierarchical approval can follow. The process of investigation, often involving theological commissions, can take some considerable time.
The Church is in a very difficult position with regard to apparitions. What if an important message for mankind is missed through ignoring them? On the other hand, what if the alleged apparition is false? The Church will not approve an apparition lightly, but if She does so, it means that there is sound evidence for the happening, and that nothing contrary to faith and morals has been found. That is, it accords with that found in scripture or tradition and accords, therefore, with Church teaching.
The credibility of the seers; the recipients of an apparition, is also carefully investigated. Approval means that an apparition has been declared “worthy of belief ” it does not go beyond that.
And an apparition/revelation is always seen by the Church in the context of the public/private revelation divide. The Church has always, and necessarily, declared that public revelation (scripture) was concluded at the end of the Apostolic age of the New Testament and that there will be no more public revelation until the Second Coming of Christ. Public revelation is the foundation of Church doctrine and anything outside of public revelation (outside of scripture) even when approved; is automatically classed as being a private revelation. And Church teaching is that there is no obligation on the faithful to believe in a private revelation. But, to emphasize, Church approval does mean that an apparition is declared “worthy of belief.”