"How does Jesus Carry on the ministry of exorcism in His Church?"

ANSWER!

Jesus Himself acts whenever an exorcist uses His Name to drive out demons; namely, the exorcist acts in persona Christi, Ordination confers a spiritual power and mandate to bless, and some priests are given special permission from the bishop to perform in a direct way the ministry of exorcism, which is a unique type of blessing reserved for those most oppressed by the power of evil.

While the performance of an exorcism is limited to priests, the Church's power to combat evil is not.  Anyone who is a baptized Christian has authority to fight evil in whatever way is appropriate to their state in life.  Most Christians do not recognize that Christ bestowed this authority upon them and therefore don't use that authority to confront the evils they face in their daily lives.  In most cases, Christians can fight evil by invoking the Name of Jesus against any manifestation of evil and by binding and rebuking it.  If they encounter truly serious cases of demonic possession, they can refer these to the ordained ministers.

By Rev Thomas J. Euteneuer "Exorcism and the Church Militant" page 10.