Babies who die before baptism according to RC

As regards children who have died without Baptism, the Church can only entrust them to the mercy of God, as she does in her funeral rites for them. Indeed, the great mercy of God who desires that all men should be saved, and Jesus’ tenderness toward children which caused him to say: “Let the children come to me, do not hinder them,” allow us to hope that there is a way of salvation for children who have died without Baptism. All the more urgent is the Church’s call not to prevent little children coming to Christ through the gift of holy Baptism. (CCC, 1261)

see also: http://nannykim-catholicconsiderations.blogspot.com/2013/04/unbaptized-babies-who-die.html

1037 God predestines no one to go to hell; for this, a willful turning away from God (a mortal sin) is necessary, and persistence in it until the end. In the Eucharistic liturgy and in the daily prayers of her faithful, the Church implores the mercy of God, who does not want "any to perish, but all to come to repentance":

and this http://www.vatican.va/roman_curia/congregations/cfaith/cti_documents/rc_con_cfaith_doc_20070419_un-baptised-infants_en.html  Here is just a part of the intro:

INTERNATIONAL THEOLOGICAL COMMISSION

THE HOPE OF SALVATION FOR INFANTS
WHO DIE WITHOUT BEING BAPTISED*

The International Theological Commission has studied the question of the fate of un-baptised infants, bearing in mind the principle of the “hierarchy of truths” and the other theological principles of the universal salvific will of God, the unicity and insuperability of the mediation of Christ, the sacramentality of the Church in the order of salvation, and the reality of Original Sin. In the contemporary context of cultural relativism and religious pluralism the number of non-baptized infants has grown considerably, and therefore the reflection on the possibility of salvation for these infants has become urgent. The Church is conscious that this salvation is attainable only in Christ through the Spirit. But the Church, as mother and teacher, cannot fail to reflect upon the fate of all men, created in the image of God, and in a more particular way on the fate of the weakest members of the human family and those who are not yet able to use their reason and freedom.
It is clear that the traditional teaching on this topic has concentrated on the theory of limbo, understood as a state which includes the souls of infants who die subject to original sin and without baptism, and who, therefore, neither merit the beatific vision, nor yet are subjected to any punishment, because they are not guilty of any personal sin. This theory, elaborated by theologians beginning in the Middle Ages, never entered into the dogmatic definitions of the Magisterium, even if that same Magisterium did at times mention the theory in its ordinary teaching up until the Second Vatican Council. It remains therefore a possible theological hypothesis. However, in the Catechism of the Catholic Church (1992), the theory of limbo is not mentioned. Rather, the Catechism teaches that infants who die without baptism are entrusted by the Church to the mercy of God, as is shown in the specific funeral rite for such children. The principle that God desires the salvation of all people gives rise to the hope that there is a path to salvation for infants who die without baptism (cf.CCC, 1261), and therefore also to the theological desire to find a coherent and logical connection between the diverse affirmations of the Catholic faith: the universal salvific will of God; the unicity of the mediation of Christ; the necessity of baptism for salvation; the universal action of grace in relation to the sacraments; the link between original sin and the deprivation of the beatific vision; the creation of man “in Christ”.

The conclusion of this study is that there are theological and liturgical reasons to hope that infants who die without baptism may be saved and brought into eternal happiness, even if there is not an explicit teaching on this question found in Revelation. However, none of the considerations proposed in this text to motivate a new approach to the question may be used to negate the necessity of baptism, nor to delay the conferral of the sacrament. Rather, there are reasons to hope that God will save these infants precisely because it was not possible to do for them that what would have been most desirable— to baptize them in the faith of the Church and incorporate them visibly into the Body of Christ.

Finally, an observation on the methodology of the text is necessary. The treatment of this theme must be placed within the historical development of the faith. According to Dei Verbum 8, the factors that contribute to this development are the reflection and the study of the faithful, the experience of spiritual things, and the teaching of the Magisterium. When the question of infants who die without baptism was first taken up in the history of Christian thought, it is possible that the doctrinal nature of the question or its implications were not fully understood. Only when seen in light of the historical development of theology over the course of time until Vatican II does this specific question find its proper context within Catholic doctrine. Only in this way - and observing the principle of the hierarchy of truths mentioned in the Decree of the Second Vatican Council Unitatis redintegratio (#11) – the topic can be reconsidered explicitly under the global horizon of the faith of the Church. This Document, from the point of view of speculative theology as well as from the practical and pastoral perspective, constitutes for a useful and timely mean for deepening our understanding this problem, which is not only a matter of doctrine, but also of pastoral priority in the modern era.