What is the basis for the practice of Intinction (the dipping of the bread into wine during the Eucharist)?

There's an article here that covers the history of intinction quite extensively. (I apologize that it's a PDF reference. It's the best article I could find.)

The article starts out like this:

The common wisdom among opponents of intinction is that it arose after the doctrine of transubstantiation, and was a method of preventing Christ's blood from spilling, and that it is associated primarily with Roman Catholicism. Although there are elements of truth in this assessment, the true history of the practice is a fair bit more complicated than that.

In summary, as far as the history goes, there's no consensus on why the practice as started. The earliest mention of the practice was in the writings of Julius I in 340 A.D.

In the context of rejecting several irregular practices regarding the Eucharist, he states the following: But their practice of giving the people intincted Eucharist for the fulfillment of communion is not received from the gospel witness, where, when he gave the apostles his body and blood, giving the bread separately and the chalice separately is recorded

There is no mention of the reason for the practice, just that it occurred.

Further, the article states:

It is not clear how or why intinction first was introduced into the church. According to William Freestone, the possibility exists that it was a convenience directed towards the administration of the Eucharist to the sick, making the bread easier to swallow. However, this was usually not our modern practice of intincting the bread into the wine, but rather of dipping bread into unconsecrated liquid. According to the Ohio Presbytery Report, the first mention of it is connected to paedocommunion, to make the bread easier to swallow by an infant. Freestone thinks it more likely that the practice originated from the fear of accident, and then passed over into communion of the sick

The remainder of the paper goes on to document various historical disputes over the practice, some of which are quite fiery. The dispute isn't really on-topic to the question, but it is worth a read.

In the Catholic Church, intinction seems to be something that is not meant to be common. I've found several articles (like this one) that state that it " is permitted when done according to Church directives". It also states "The Church provides that when Communion is given by intinction, it must be planned beforehand.", indicating that it's not something to be taken lightly. But nowhere does it outline why it should be given, and in what cases.

The only article I could find that had any sort of official support for there answer was this one.

With reference to the previous section on the theology of intinction (i.e., responses to Questions 2 and 3), this study committee found strong, compelling biblical and theological support for the practice of partaking of the elements separately – of “eating” AND of “drinking”. Therefore, intinction is out of accord with Scripture.

Practical considerations appear to be the primary reasons for intinction. During this study, the committee heard a variety of pragmatic reasons for the practice, including: (a) it is one of the “touch points” for a more meaningful worship, (b) it saves time, (c) it takes longer, (d) it may better appeal to those who come from church traditions that practice intinction (e.g., Catholic, Orthodox), (e) it avoids the Congregationalist practice of distributing the elements in the pews, (f) it better enables communion in the battlefield, and (g) it is practiced in the PCA.

Again, it points to convenience, or practicality as a reason.

From the Anglican Church:

Resolutions from 1948

Resolution 118

Administration of Holy Communion

The Conference holds that administration from a common chalice, being scriptural and having a spiritual meaning of great value, should continue to be the normal method of administration in the Anglican Communion; but is of opinion that there is no objection to administration of both kinds by the method of intinction where conditions require it, and that any part of the Anglican Communion by provincial regulation according to its own constitutional procedure has liberty to sanction administration by intinction as an optional alternative to the traditional method, and that the methods of intinction to be adopted or permitted should not be left to the discretion of individual priests.

The only guideline appears to be "where conditions require it". Implying some special set of circumstances where the traditional method is inconvenient or impractical.

I did manage to find one specific example that lists a specific reason:

Q. Why does St. Peter’s intinction, rather than using the cup like other parishes?

A. There are several reasons. We have had some difficulty in the past getting a sufficient number of EMHC. And even if we had them, numerous EMHC sometimes make for a rather cumbersome crowd in the Sanctuary at Communion time. There is the additional problem of preparing the proper amount of wine for the Offertory. When the Precious Blood is then distributed, we either run out before all have received, or have much remaining, which then must be consumed by the Priest and/or EMHC. Many people also refrain from receiving the Precious Blood of the Lord from the Cup because of hygiene concerns. Intinction eliminates these problems.

So (and I'm repeating myself) it is a matter of practicality for them as well.