Alice,
Generally, the persons bringing up the gifts should be Catholic but ultimately the priest or pastor will decide policy for the parish in practice. I'm a Music Director at my parish. I play at over one hundreds funerals a year and there are occasions where, in my parish, the pastor allows family members (non-Catholics or non-practicing Catholics) to bring up the gifts.
It is a different matter with respect to the reception of Holy Communion. That is reserved for those who make a full profession of faith in the Catholic faith and whose conscience is clear — having confessed all serious sin.
This sacrament is a sign of unity amongst the Church so it would become a false sign if participants were not adherents to Catholicism and didn't practice the faith. (Consider that the word sacrament means oath.)
Catholics and non-Catholics alike may attend Mass, but only properly disposed Catholics may receive Holy Communion for the above reasons. In some parishes, as ours, the pastor may invite others to receive a blessing at Communion time.
Peace,
Bob Kirby
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