Both armies didn't pull absolutely all of their cavalry from the flanks and pour it into the center as they still needed them to guard exits and roads of escape and communication. The reason why the large amount of cavalry was concentrated in the middle was because of the British charge at the French guns, the French counter, and then Ney's large cavalry assault against the British line. Wellington told the army to "retire 100 paces" at which point they rounded over the hill and disappeared. Ney believed Wellington was retreating and ordered forward some cavalry to pursue them. Somewhere in his shouts of orders he increased his quantity of cavalry to include something around 67 squadrons. The British infantry countered this by forming squares and whatever was left of the British Union Brigade (from the earlier assault on the French infantry columns) was committed By Uxbridge to help the squares.

Source is Waterloo: Day of Battle by David Armine

Heavy and Line Cavalry is of very little use on the flanks; that is for light cavalry. The definition and classification of the three change greatly from say, 1750, to 1812. Dragoons become line and 'heavy' cavalry, for example, in the 1800s; and Hussars and like light cavalry range further and further from the army, extending flank protection to a more operational level.

For the Alliance, they were on the tactical defensive, and Wellington had picked perfect defensive terrain. There was no need to over-commit cavalry to the flanks; heavier cavalry, such as the now-famous Scots Greys, stayed with the rest of the heavier mounts in a central reserve. Its a standard principle of concentration of force, and always keeping a reserve in the most flexible position to react to success or disaster - usually, and in this case, the rear-center of a battle line.

Imagine it this way: Does a modern army keep slow-moving, hard-hitting, gas-guzzling armor units on the flanks? No. Heavy Cavalry needs to be kept fresh and in the closest position to hit the enemy line as possible; you don't want to 'blow' a charge. By keeping the heavy cavalry in the center, both the French, and the British, were in a position to attempt to exploit any successes, or use cavalry as a reserve to buy time and reform after a disaster. Your question can be answered simply with the maxim of concentration of force; you strike with a closed fist, not an open hand. I go into a bit more detail here on how Napoleonic cavalry was used in conjunction with other arms in a typical army of the time.

I thought cavalry was supposed to be on the flanks? Was there a change in cavalry doctrine in Europe? If there was when did this take place and why?

They used to be, yes. This was largely due to the primacy of infantry over all other arms in terms of mobility and hitting power. Artillery was theoretically the king of the battlefield, but had not yet reached a rate of production or quality of production to make them rapidly moving. Let's put it this way...

Here is an image of the Battle of Prestonpans; from the last Jacobite rebellion. Notice the British cavalry is on the flanks? Battle lines were small, battles were generally much smaller, armies were much smaller, etc. Cavalry didn't often have the numbers to be a decisive hitting force, and were used for screening and flank protection thusly.

Here by comparison, is the Battle of Friedland, during the afternoon situation. What are you looking at here? If you're not familiar with NATO unit designations, a single slanted stripe represents cavalry. So both armies has a strong force of cavalry opposite one another on the top of battlefield; guarding the flanks, nothing has changed there; Friedland was an open ended battlefield and flank protection was absolutely necessary. But the French still have one, reinforced heavy cavalry division smack dab in the center, with Lannes Corps. ("Nansouty's command") as a potential exploitation force, reserve force, or attack of opportunity.

Why are they able to do both? Simple: Numbers. Cavalry arms had been growing in size and skill across all armies, and some of the finest Cavalry arms by the end of the Napoleonic Wars; the British, French and Austrian arms, were very much more egalitarian afterwards. Horses bred by the state, enlisted men from the general population, etc. Britain and France proved that Cavalry corps made up of 'commoners' could still be well-handled and skilled horsemen with proper organization and leadership; and thus cavalry arms exploded in size. They got large enough to be concentrated in divisions and entire corps, amounts of horsemen not really seen on a battlefield in the 1700s; or at the very least, not concentrated in such numbers.

Edit:

And for that matter if Napoleon was trying to drive Wellington away from Blucher why not concentrate his cavalry on his right and make that where the hammer falls? When did cavalry supporting infantry in melee and preventing the infantry from routing start taking precedence to the older procedure of concentrated on the flanks to secure one's flank against enemy cavalry and hopefully defeating it and turning on the enemy infantry in the center? And lastly, sources for these reasons for deployment you mention?

Firstly, Napoleon had attempted to do that strategically. As far as he was concerned, Grouchy had achieved this and was expecting Grouchy, not Blucher, to arrive on the field first. This is where the seeds of defeat were planted for the French.

As for the second question, I've already touched upon that I believe, earlier in the question. Small forces of cavalry 'turning in' on the flanks rarely accomplished anything on a Napoleonic battlefield; well supported squares swept by artillery fire could usually halt, or funnel, even a Brigade sized force of cavalry. A Cavalry strike needed to overwhelm defensive artillery, a hammer blow, not a slap, is necessary; and its far too risky to concentrate the numbers necessary on a flank, while denuding yourself of any tactical flexibility.

Flexibility, flexibility, flexibility. The keyword here; and the reason why both armies opted for a central reserve rather than a staggered one. Its easier to trot a central reserve of cavalry to a flank in preparation for supporting it, from a central position, than it is to run a cavalry from a stacked flank all the way to the opposite one in case of unforseen developments. Neither General at Waterloo was stupid enough to do that! I think you over-estimate just how rapidly events could actually develop on a battlefield...

It's a very interesting question and I dont really recall ever having read much about it. It seems most sources I have read on Waterloo do not seem to give much explanation; as if it was all obvious.

As far as doctrine goes, I think for most of the Napoleonic wars the idea was that a combined arms attack was best and the idea was that large enemy infantry formations would be forced to form squares and then artillery could hit bombard these. Presumably infantry would also need to be involved. You have to read Brent Nosworthy's book for more on this, but you will find this general idea in many sources.

In fact something like this did happen in the latter stages of the battle, when Wellington's formations east of La Haye sainte were decimated by artillery; while Ney's cavalry had been or was in the process of attacking them. So there was some sort of doctrine to support this concept of cavalry acting in concert with artillery/infantry.

As for Waterloo, no I dont think this was an unusual case. You can study maps of many Napoleonic battles and you will see cavalry often deployed directly behind infantry. At the Ligny battle, two days previous, there is a large Prussian formation, under Roeder right in the middle of their lines. The french cav. is more traditional, on the flanks but some of it is in the center.

At Borodino 1812 there are a number of Russian cav. units deployed just as Napoleons at Waterloo; similar for the French. I think in this case, cavalry was meant to follow up successful infantry attacks and sweep demoralized survivors off the field. At Leipzig 1813, you see some of the Allied cavalry more in between infantry units so maybe it is acting more as a link between infantry units. Note Leipzig, both in size and numbers is vastly larger battle, the largest of Napoleonic so maybe there is some details missing. I try to give some examples that are close in time to Waterloo, here.

So one thing I am pretty certain, it is not an unusual position for cavalry.

As to the purpose, I think in Napoleon's case, the idea is often mentioned that he thought he would sweep the English off the battlefield and then quickly capture Brussels. If the idea was to capture the city quickly, he wanted his cavalry close to the main road where they could move right passed the survivors and quickly into the city. Obvious, this plan was over ambitious and Nap. seems to be extremely overconfident.

Also for the French one should take into account that there were only a couple of bridges over the Dyle river that led to the battlefield, and the soldiers were more or less chanelled into the battle. Also there were heavy rains the day before and the formations may have been positioned simply out of necessity, they did not have the time and energy to work around the flanks, so they lined up as they arrived.

Also recall that this campaign lasted only about 3 days, there were units that were pretty much on the march the entire time; for the French they were pretty much tied to the Charleroi Brussells turnpike that was their one main road and so this may also be a factor. I.e. they were channeled along the main road simply because the campaign was waged very quickly, they did not have time for outflanking.

Also the terrain at Waterloo: the ground was very much rolling; even in places were that appears flat on battle maps there is much rolling terrain. Especially on Nap's right, like from Plancenoit to Frischermont it is very rolling and perhaps difficult for cavalry. The area around Placenoit is very rugged because of the tributary stream in that area. I do not know, but I would guess this whole area may have been a quagmire. Not sure about Nap.'s left, the area west of Hougomont. I think some of the horse artillery was used here against the Hougomont complex, however I think also mud limited their movements.

Note also: the battlefield was also dominated by those reinforced farmhouses Hougomont, Frischermont, La Haye St, etc. (some of these were actually built up because Louis XIVth wanted reinforced buildings on these highways) Both of these factors (fortified houses/terrain) may have made traditional cavalry charges on the flanks out of the question. Also the mud, I think bears constant repeating, some of the areas must have been very soaked.

For the British, I think it is even more interesting question: Wellington's position here is similar to how he usually fought in Spain: in a dogged defensive position with perhaps a counter attack at the right moment. I dont think he had much cavalry to work with in Spain, and you usually see it on the flanks there. In any case, the cavalry on his extreme left: Vivien, Vandeleur, Vincke, etc I think these were lighter cavalry and were supposed to be some sort of liason with Bluchers forces.

The heavy cavalry was in the center; and I suppose this must have been part of this plan: to counter attack at the right moment with heavy cavalry. It worked to some degree.

My conclusion, the terrain and the weather, more or less channeled the cavalry into a role where it had to support infantry in making headlong attacks here. There was very little dry ground or flat ground on which to make the typical flanking attacks that cavalry is known for. I dont think this was at all unusual.

EDIT: It seems to me that the perhaps the only good place for cavalry to fight was just off either side of the Brussells Charleroi Road. Ney's charges in later afternoon, west of this road, and the British heavy cavalry earlier in the day, east of the road. As I recall, at one point the french cavalry early in the battle went over the crest of Wellington's ridge, near the sandpit, east of La Haye St. but they did so in more or less columns instead of waves. This must have had something to do with the heavy amount of mud that is mentioned all the time. So again even cavalry was probably limited to using the few good roads, at least to get into position, so this may have a lot to do with it.

Sources: my most recent book is the one by David Hamilton Williams, Waterloo New perspectives. It has been criticized for its conclusions/opinions but there is a lot of factual information there. David Chandler: Campaigns of Napoleon is a standard, but not much on tactics. Brent Nosworthy: WIth Musket, Cannon and Sword is good on tactics, not Waterloo in particular.