The key issues seem to be in the Code of Canon Law, Chapter VI, canon 1125, which says in part:
1/ the Catholic party is to declare that he or she is prepared to remove dangers of defecting from the faith and is to make a sincere promise to do all in his or her power so that all offspring are baptized and brought up in the Catholic Church;
2/ the other party is to be informed at an appropriate time about the promises which the Catholic party is to make, in such a way that it is certain that he or she is truly aware of the promise and obligation of the Catholic party;
3/ both parties are to be instructed about the purposes and essential properties of marriage which neither of the contracting parties is to exclude.
A theme common to all these clauses is an attempt to maintain the Catholic partner and all future children as members of the Catholic faith, whether out of concern for their souls or concern for Church membership. To ensure that the Catholic faith is seen by both partners as paramount and that they regard their marriage as an exclusively Catholic sacrament, canon 1127, clause 3, says:
It is forbidden to have another religious celebration of the same marriage to give or renew matrimonial consent before or after the canonical celebration according to the norm of §1. Likewise, there is not to be a religious celebration in which the Catholic who is assisting and a non-Catholic minister together, using their own rites, ask for the consent of the parties.
Until around the middle of the twentieth century, the Catholic Church tended to insist the non-Catholic partner convert to Catholicism and only permitted mixed marriages as a last resort, in cases where conversion was refused and the Catholic partner was at risk of marrying outside the Church. Which brings us to the second part of the question - the fact that Catholics were increasingly prepared to enter into mixed marriages, with or without Church approval, brought about a relaxation of this impediment. Now, there are many Catholics who don't think this is an issue and are willing to make the required undertakings merely as a formality, or to marry outside the Church. Marrying outside the Church might mean the Catholic is no longer able to receive communion, but worldwide statistics from 2004 show that in many nominally Catholic countries, a minority of the population even goes to mass every week.
Apart from dispensation to marry outside the faith, subject to the above requirements, it is not possible to provide evidence against this 'belief', because those requirements are not a belief but Church doctrine. Something that is Church doctrine is its own proof.