You need to understand that RCIA is not a class where you cover certain material, perhaps take a test, and get a diploma.
The most basic question is what is RCIA meant to accomplish? See paragraph 75 of the Rite:
- The catechumenate is an extended period during which the candidates are given suitable pastoral formation and guidance, aimed at training them in the Christian life. In this way, the dispositions manifested at their acceptance into the catechumenate are brought to maturity. This is achieved in four ways.
1. A suitable catechesis is provided by priests or deacons, or by catechists and others of the faithful, planned to be gradual and complete in coverage, accommodated to the liturgical year and solidly supported by celebrations of the word. This catechesis leads the catechumens not only to an appropriate acquaintance with dogmas and precepts but also to a profound sense of the mystery of salvation in which they desire to participate.
- As they become familiar with the Christian way of life and are helped by the example and support of sponsors, godparents, and the entire Christian community, the catechumens learn to turn more readily to God in prayer, to bear witness to the faith, and in all things to set their hopes set on Christ, to follow supernatural inspiration n their deeds and to practice love of neighbor, even at the cost of self-renunciation. Thus formed, "the newly converted set out on a spiritual journey. Already sharing through faith in the mystery of Christ’s death and resurrection, they pass from the old to a new nature made perfect in Christ. Since this transition brings with it a progressive change of outlook and conduct, it should become manifest by means of its social consequences and it should develop gradually during the period of the catechumenate. Since the Lord in whom they believe is a sign of contradiction, the newly converted often experience divisions and separations, but they also taste the joy that God gives without measure.
- The Church, like a mother, helps the catechumens on their journey by means of suitable liturgical rites, which purify the catechumens little by little and strengthen them with God’s blessing. Celebrations of the word of God are arranged for their benefit, and at Mass they may also take part with the faithful in the liturgy of the word, thus better preparing themselves for their eventual participation in the liturgy of the eucharist. Ordinarily, however, when they are present in the assembly of the faithful they should be kindly dismissed before the liturgy of the eucharist begins (unless their dismissal would present practical or pastoral problems). For they must await their baptism, which will join them to God’s priestly people and empower them to participate in Christ’s new worship.
- Since the Church’s life is apostolic, catechumens should also learn how to work actively with others to spread the Gospel and build up the Church by the witness of their lives and by professing their faith.
Read through that and it is very clear that RCIA is not a class, it’s an introduction to a way of life. Going from one parish to another to another may be possible, but it’s definitely not ideal.
Perhaps it would be best to wait until your life is more stable and you’re not moving around every few months. For now, you can certainly attend Mass in whatever place you’re living. Take advantage of anything the parish in your area has to offer in the way of lectures, short-term classes, service activities, etc.
Or, depending on how far these various moves will take you, perhaps you could select one parish as your home and commit to participating in RCIA at that parish. You might have to drive a few miles to stay at the same parish, but it would give you some stability during what sounds like a somewhat chaotic period of time.
