In this video by the Institute of Christ the King Sovereign Priest, we can see brief moments of the procession into the church. At the front of the procession with the cardinal (at 5:04), there appears to be a man dressed in archaic secular clothes holding a red hat. It seems obvious this is related to the dignity of a cardinal, but why is he dressed that way?
Lastly, in the same video we see at minute 2:58, a man wearing, again, archaic secular dress wielding a pike. Why is he there?