Transubstantiation and John 6:27

The Catholic Study Bible's note on this verse cross references Jesus' dialogue with the Woman at the Well, who He said would be better off drinking Life-Giving Water. He said nearly the exact same thing to her. "Why waste time drawing water that makes you thirsty again, try Life-Giving Water instead." And later the crowd clamored for Jesus to give them this bread, exactly the same way the Woman asked Jesus to give her that water.

However, the Church doesn't mention this passage in the Eucharistic Prayer the way she instructs Priests to say "do this in remembrance of Me". The bread and wine is the people's sacrifice during Mass. It is turned into the Body and Blood, Soul and Divinity of Jesus - so long as it appears as bread, otherwise you start thinking that all the molecules that make up the bread are Jesus and then they're absorbed into our body and/or "passed out into the latrine". This is not Catholic teaching. We believe that when the Eucharist is consumed, it stays with you as long as it retains the appearance of bread - I've heard this is about 10-15 minutes.

On the other hand, to dispose of the Consecrated Host (and this part might answer your question) one must reverently dissolve the bread in water until it no longer resembles bread.

If not consumed the Host is allowed to dissolve in water, such as the lavabo bowl, and when it has completely disintegrated the contents can be poured into the sacrarium (or in its absence buried).

EWTN Q&A

A similar set of rules applies to Consecrated Blood.

As a child, I used to think that the hosts would never run out, like Jesus' miracle of loaves and fishes, until I started paying attention to what was actually happening, which is pretty clever because consecrated hosts are not usually left to rot. Although it is certain that they could, the likelihood of one being left long enough is slim as Masses are offered every day and usually an appropriate amount of hosts are consecrated.

Can. 939 Consecrated hosts in a quantity sufficient for the needs of the faithful are to be kept in a pyx or small vessel; they are to be renewed frequently and the older hosts consumed properly.

It's explicit in the text, not in Catholic theology only, that "the bread I will give" is, quote, "my flesh." (v. 51).

What makes you think Jesus' flesh will perish? It is spiritual food, not only physical food. As such, it doesn't perish, but remains in us.

As for v. 51 see the institution of the Eucharist where He says of the bread, not using the pronoun 'this' for 'bread' (which is masculine) but the neuter pronoun (corresponding to 'body'): "This is My body." Unless 1) He was playing charades, or 2) lying, it was literally and really made His flesh.

In addition, you have to consider why this one teaching scandalized His disciples, even almost making His Apostles leave Him. They didn't say 'this is a badly chosen saying,' (which it would be if it weren't true) but 'this is a difficult saying, who can hear it?' Who can't hear or won't listen to the doctrine that Christ dwells within us? Literally no one. But who wants to hear that we eat the Bread of Life? Few indeed. Because they are carnally minded, not spiritually minded. His audience were thinking of physical nourshment, whereas He is the Bread of Life in a greater way than mere physical sustenance, but is the Bread spoken of in the Our Father: our daily Bread, our very life. Now Jesus could have taken back what He said; instead, He used more vivid words, and repeated what He said.

You should also consider that St. Paul said eating unworthily the Eucharist leads to one's damnation because they haven't recognized the body of the Lord for what it is, and are guilty of the body and blood of the Lord. This is impossible if the Eucharist is not the body and blood of the Lord.

Lastly, what does our Lord mean by the words 'Do this' in the institution of the Eucharist, except to hold the bread in the hands and say what He said, 'This is my body,' as priests have for 2000 years?

St. Thomas Aquinas explains that verse in his commentary on St. John's Gospel ch. 6 (my emphasis):

895. The power of this food is seen in the fact that it does not perish. In this respect we should point out that material things are likenesses of spiritual things, since they are caused and produced by them; and consequently they resemble spiritual things in some way. Now just as the body is sustained by food, so that which sustains the spirit is called its food, whatever it might be. The food that sustains the body is perishable, since it is converted into the nature of the body; but the food that sustains the spirit is not perishable, because it is not converted into the spirit; rather, the spirit is converted into its food. Hence Augustine says in his Confessions: “I am the food of the great; grow and you will eat me. But you will not change me into yourself, as you do bodily food, but you will be changed into me.”

Related:

• Summa Theologica III q. 77 a. 4: "Whether the sacramental species can be corrupted [i.e., decompose]?"

^{cf. this answer to this question regarding the nutritional aspect of Holy Eucharist}