"And how does this happen to me, that the mother of my Lord should come to me?"
Jesus warns us about going to the front of the table just in case someone else more important comes in (Luke 14:7-11). Elizabeth seems to be the inspiration for the correct behavior that Jesus speaks about. She immediately defers to Mary. Certainly it usually would have been her, the older woman, who would have come to the aid of the younger woman in her pregnancy, and yet as the one further along, Mary rushes to aid her.
Elizabeth experiences only joy; her own infant leaps in her womb. Both her and John within her defer to Mary and Jesus within her. Elizabeth does not boast about her late in life pregnancy, does not make herself and her child the center of attention - does not take the seat at the head of the table. She is not embarrassed when Mary arrives.
I am certain that her pride would have swelled when she was eventually with child, to no longer feel the shame of being thought to be barren. The women of the time must have fawned over her and made her feel important. But it seems she is much more demure and much more willing to not have everything be about her.
This understanding she must have instilled in her son John. His actions and words show that while wildly popular and spiritually powerful he did not sit at the head of the table but took his proper place such that when Jesus arrived he could say:
"Behold, the Lamb of God, who takes away the sin of the world. He is the one of whom I said, ‘A man is coming after me who ranks ahead of me because he existed before me.’ I did not know him, but the reason why I came baptizing with water was that he might be made known to Israel.” (John 1:29b-31) And then Jesus could say of John: "Amen, I say to you, among those born of women there has been none greater than John the Baptist; yet the least in the kingdom of heaven is greater than he." (Matthew 11:11)
When Elizabeth heard Mary’s greeting, the infant leaped in her womb, and Elizabeth, filled with the holy Spirit, cried out in a loud voice and said, “Most blessed are you among women, and blessed is the fruit of your womb. And how does this happen to me, that the mother of my Lord should come to me? For at the moment the sound of your greeting reached my ears, the infant in my womb leaped for joy. Blessed are you who believed that what was spoken to you by the Lord would be fulfilled.”
-- Luke 1:41-45