The tradition of the father giving away his daughter harkens back to the days when children, daughters in particular, were considered to be a possession of their fathers. Marriages then were, for the most part, arranged by the two sets of parents. Because a daughter belonged to her father, he had the legal right to give her child to the groom, more often than not, for a price.
Today a father giving away his daughter is a symbol of his blessing of the marriage. It is also a public display which demonstrates his confidence in "handing over" the responsibilities involved with taking care of his daughter to his son-in-law to be. It acts a symbol of the bride's progression from childhood to adulthood.
There is probably very little chance that this tradition will fall out of favor, but, there are some objections being raised by the Catholic Church. Some members of the Church believe that the tradition unnecessarily lengthens the procession and makes it into something of a fashion show. The feeling is that the Church may wish to focus the ceremony more from being bride-centered to being bride and groom centered. Eliminating the "giving away" may be a step in that direction.
In traditional Jewish wedding ceremonies, a more equal approach that has both the bride and the groom proceeding down the aisle accompanied by their respective sets of parents. Another alternative that is being suggested is that the groom walk down the aisle unescorted and that he be followed by the bride and her parents.
Regardless of its roots, the tradition of walking down the aisle with their dads is one that, for the most part, continues to be cherished by fathers and daughters alike.
As far as how to adapt the ritual when couples are divorced, the method depends very much on the relationship between the birdie and her father and, if her mother has remarried, with her stepfather. The options are that she walks down the aisle with her father only, with her stepfather only, with both, or with neither, selecting an older brother, perhaps, to stand it. It's a very personal decision which needs to be approached on a case by case bases.