The two links below are examples of many, many Roman Catholic resources explaining the origin, significance, and liturgical format of foot washing (The Mandatum) which takes place during Maundy Thursday services.
The biblical text undergirding this practice is in John 13:
Now before the Feast of the Passover, when Jesus knew that his hour had come to depart out of this world to the Father, having loved his own who were in the world, he loved them to the end. During supper, when the devil had already put it into the heart of Judas Iscariot, Simon's son, to betray him, Jesus, knowing that the Father had given all things into his hands, and that he had come from God and was going back to God, rose from supper. He laid aside his outer garments, and taking a towel, tied it around his waist. Then he poured water into a basin and began to wash the disciples' feet and to wipe them with the towel that was wrapped around him. He came to Simon Peter, who said to him, “Lord, do you wash my feet?” Jesus answered him, “What I am doing you do not understand now, but afterward you will understand.” Peter said to him, “You shall never wash my feet.” Jesus answered him, “If I do not wash you, you have no share with me.” Simon Peter said to him, “Lord, not my feet only but also my hands and my head!” Jesus said to him, “The one who has bathed does not need to wash, except for his feet, but is completely clean. And you are clean, but not every one of you.” For he knew who was to betray him; that was why he said, “Not all of you are clean.” When he had washed their feet and put on his outer garments and resumed his place, he said to them, “Do you understand what I have done to you? You call me Teacher and Lord, and you are right, for so I am. If I then, your Lord and Teacher, have washed your feet, you also ought to wash one another's feet. For I have given you an example, that you also should do just as I have done to you. Truly, truly, I say to you, a servant is not greater than his master, nor is a messenger greater than the one who sent him. If you know these things, blessed are you if you do them. - John 13:1-17
I will pull excerpts from this passage as I try to clarify my questions (asked of Roman Catholicism).
a) "For I have given you an example, that you also should do just as I have done to you." Since this took place during the same Passover meal that Christ eagerly desired to eat with His disciples and during which the Lord's Supper was instituted, and since the act appears preparatory to that first Eucharist, why does the Roman Catholic Church only incorporate this into one liturgy each year while the Eucharist takes place during each liturgy?
b) "If I then, your Lord and Teacher, have washed your feet, you also ought to wash one another's feet." Given that Jesus washed the feet of everyone there (apparently even Judas) and given that Jesus commanded them to likewise wash one another's feet, why does the Roman Catholic celebration of this event involve only the principle celebrant (Bishop, Priest, or those he designates) doing the washing to only a certain number of chosen congregants? An ancillary question here is; Are Catholic parishioners likewise encouraged to wash one another's feet going forward?
c) "If you know these things, blessed are you if you do them." I guess this question is more overarching. Given the tight Scriptural linkage (not just by proximity but by intentionality) between the foot washing (I have given you an example that you should do) and the Lord's Supper (Do this in remembrance of me), why does Roman Catholicism hold the latter as the literal and indispensable centerpiece of each liturgy while the former is only remembered yearly and symbolically?