If God uses people like this to draw others to Him, but they do not have any God-given apostolic succession authority as leaders, then how does that work?
God can work in the lives of any and all who desire Him, even if they are misguided. That being said, just because people are zealous, does not mean they are necessarily being moved by the Holy Spirit.
Jesus said to His disciples when they were upset about non-Christian Jews casting out demons and performing miracles that they were not against Him, so to let them be.
Indeed. It is not our job to pull out the weeds. Jesus said to let the weeds and the wheat grow together until the harvest, and the angels will do the sorting.
But then Jesus says that He will separate the sheep from the goats and even those who performed miracles and such He will not recognize with His sheep.
What is the Catholic response to this?
Catholics need to be focused upon keeping themselves in the state of Grace. We each of us have one primary responsibility, that is to get our own souls to heaven. After that, we are responsible for those God has given into our care (such as spouses and children). There are many ways to tell if one is in a state of grace. One of these is by observing the fruit:
“…the fruit of the Spirit is love, joy, peace, patience, kindness, generosity, faithfulness…” Galatians 5:22
We can also measure by the degree of love we see:
“Love is patient, love is kind. It does not envy, it does not boast, it is not proud. It does not dishonor others, it is not self-seeking, it is not easily angered, it keeps no record of wrongs. Love does not delight in evil but rejoices with the truth. It always protects, always trusts, always hopes, always perseveres. Love never fails.” I Cor. 13: 4-8