Based on the New Testament. The Holy Spirit was given to the Apostles, along with the power to bind and loosen and with the keys specifically given to +Peter. It is consistent with the Bible, as you say, that the Holy Spirit was passed on by those who received it to their successors.
If you believe that the Holy Spirit never leaves the elect, then you must believe that the same Holy Spirit indwells in the Church to this day and that the Church is made up of the elect.
I'm not sure it can be said whether it is consistent or inconsistent. As far as I know, the Bible is neutral on this issue, and really doesn't mention it. I'm not familiar with scripture that says that the Spirit is passed from one human to another. Perceived authority may be passed along, but I'm not aware of how a person may pass along the Spirit Himself.
My belief is that the Spirit indwells each person at salvation. Surely knowledge and teaching are passed on, but the supernatural powers part (infallibility, forgiveness of sins, etc.) is what concerns me.
If you believe that the Holy Spirit never leaves the elect, then you must believe that the same Holy Spirit indwells in the Church to this day and that the Church is made up of the elect.
I can never speak to the election of anyone else with certainty, only myself. Certainly the Holy Spirit indwells many Catholics and Orthodox, however, I think it must also be true that there are some, who claim to be Catholics or Orthodox in faith, who are actually going to wind up in hell, JUST AS the same is true with some Southern Baptists.