Indeed it was Ollivander who introduced this notion that if a witch or wizard should "beat" another, then the victor should simply own the loser's wand and that wand should naturally respond to its new carrier.
Such was a vital issue in Deathly Hallows, especially when considering the devious plot twists centered on the Elder Wand's bloody legacy. Not to mention the rule's importance at the final resolution of the series where the loyalty of the Elder Wand, in reality, cemented Harry's victory.
Now, as was made clear in Deathly Hallows, the term "beating" really has a variety of meanings.
Let us look at the Elder Wand's bloody path and see what we can derive:
1) Death -> Antioch Peverell (Killed in his sleep)
2) Emeric "The Evil" (Died in a duel) ->
Egbert (Unkown demise)
3) Godelot (Son locked him in a cellar) ->
Hereward (Stole it)
4) Barnabas Deverill -> Loxias (Unknown demise)
5) Gregorovitch -> Grindelwald (Stole it)
6) Grindelwald (Defeated in a duel) -> Dumbledore
7) Dumbledore -> Draco Malfoy (Expelliarmus)
8) Draco Malfoy -> Harry Potter
(Stole Draco's regular wand)
When taking the above into consideration, it's obvious that the term "beating" can be interpreted rather loosely. As was proven by Dumbledore's defeat of Grindelwald in 1945, the previous owner of the Elder Wand does not necessarily need to have died for the transfer of loyalty to take effect.
Clearly, this wand of power responds to violence. It has been stolen, killed for, and betrayed for. It is a highest treasure in the wizard world - the ultimate fantasy of people young and old. And yet, it attracts those who are most vicious, people who are overcome with greed. Much like the Mirror of Erised, which allows people to see their deepest desires, the wand does not really promise greatness. The Elder Wand is notorious for bringing misfortune to its carriers going back as far as Antioch Peverell.
"The Elder wand has never been destroyed or buried, but has survived to accumulate wisdom, strength, and power far beyond the ordinary."
Luckily, unbeknownst to Dumbledore - (or was it!) - Harry saw to it that the wand was returned to his tomb.
The behavior of theeElder wand seems to shed some light on the nature of wands in general. This loyalty of wands (or lack thereof) adds a new face to their character. Perhaps, they are no more tools than the sun or the moon. While one can harness a wand's power, channel their own magic through it, perhaps, can they ever truly own it?
There's also something very important to remember: the Elder Wand is a Deathly Hallow. Whether it was indeed given by Death himself, or simply a dark creation of the eldest Peverell, the Elder Wand is altogether different from the normal, and should be judged as such. Considering its extraordinary properties, the wand likely follows the same rules of the others but to an extreamly greater extent. Meaning where a regular wand may adapt to a new carrier's sparring strategies over time as it changes hands, and slowly become more effective for that witch or wizard in a duel, the elder wand would would do the same (faster), and also protect its new posessor with techniques from masters past in a pinch.
This is some intense stuff! In any case, we can't have full understanding of wands until we unravel that exciting mystery that is Priori Incantatem.