Congratulations on asking a fascinating question that never ever crossed my mind. I hope it gets answered. Follow up: did medieval armies train during peacetime (or wartime I suppose)? Pick up heavy rocks or jog around the courtyard? Things like that.

I won't be able to provide much detail on Medieval Knights specifically, although I am able to go into some detail of battle preparations of 10th and 11th century Byzantine Roman armies, as well as their late antique predecessors.

A quick preface, I almost didn't answer this as I am in the middle of a move and most of my books are in boxes a few towns over (long story) but this hasn't been answered so I'll do my best with the sources I do have access to along with memory.

Byzantine Roman armies of the Middle Ages and also the Late Antique period were, in general, trained professional soldiers (Tagmata) serving for extended periods of time across multiple years and campaigns. There were more basic soldiers/levies (Themata) mixed in but in a general sense we have a group of career warriors well versed in all aspects of warfare. The reason this is important is we're talking about a serious fighting force, using the latest tactics, training, and technology of the times.

Our sources makes it exceedingly clear that a pitched battle (i.e. two armies lined up opposing one another in a direct engagement) is incredibly risky and should only be attempted if a positive outcome is almost assured. The Emperor Nikephoros Phokas of the 10th century provides a comprehensive account of a general's approach to battle:

It is imperative first to find out the number of the enemy host and above all what kind of equipment they have, by means of spies, deserters, and prisoners. If the enemy force far outnumbers our own both in cavalry and infantry, avoid a general engagement or close combats and strive to injure the enemy with stratagems and ambushes. The time to seek general engagements with the enemy is when, with the help of God, the enemy has fled once, twice, or three time and are crippled with and fearful, while on the other hand our host is obviously confident and their thoughts of valor have been awakened. Avoid not only an enemy force of superior strength but also one of equal strength, until the might and power of God restore and fortify the oppressed hearts and souls of our host and their resolve with His might hand and power. And when this love and compassion of God is stirred by the intercession of the all-celebrated Mother of God, when She secures Her people's victory for the third time, from that moment on they need not flinch or recoil in fear, be the enemy twice the number of our host.

Nikephoros will not engage an equal or larger enemy but avoid confrontation, ambushing and harassing, until the most opportune time.

To quote an earlier, 6th century source we can see this approach to pitched battles is largely the same:

Warfare is like hunting. Wild animals are taken by scouting, by nets, by lying in wait, by stalking, by circling around, and by other such stratagems rather than by sheer force. In waging war we should proceed in the same way, whether the enemy be many or few. To try to simply overpower the enemy in the open, hand in hand and face to face, even though you may appear to win, is an enterprise which is very risky and can result in serious harm. Apart from extreme emergency, it is ridiculous to try to gain a victory which is so costly and bring only empty glory.

My point here is the pitched battles, where both sides know there's going to be a fight at a certain time were quite rare in the Late Antique and into the Medieval periods. Small scale engagements were much more common, and in fact, small scale engagements were often the unexpected catalyst to larger battles. Often scouting or advance parties skirmishing and then escalating into a general battle. In such a situation, the larger army could be required to act immediately, with little time to prepare (including stretching and warming up).

Now that we've gone over the infrequency of pitched battles we can go into some further detail on how Byzantine Roman forces prepared for them when they did occur.

The 10th century text of the Praecepta militaria by Nikephoros Phokas, goes into some detail of the days coming up to battle although not much directly proceeding the battle. In preparation the soldiers are to begin a three-day purification through prayer, fasting and reconciliation. While some of the prayers are specifically quoted in the text, Phokas does not go into much detail on exactly what he means by a fast. It seems unlikely that any fighting force would be combat ready after three days of no food and little water but we simply don't know more. It was also important that the three days were spent in camp, with a spiritual focus. It is important that the men do not overexert themselves at this key time.

Not overexerting the troops coming up to a battle is actually a recurring theme all the way back to a 4th/early 5th century source, De re militari by Vegetius, and also the 6th century source, Strategikon by Maurice.

De re militari is actually a bit controversial for it's view on how the Romans, the subject of the work, actually fought. However, in the context of this question, it is highly relevant as De re Militari was commonly read by commanders of the Middle Ages. Vegetius comments on the importance of keeping the troops fresh:

Troops must never be engaged in a general action immediately after a long march, when the men are fatigued and the horses tired. The strength required for action is spent in the toil of the march. What can a soldier do who charges when out of breath? The ancients carefully avoided this inconvenience, but in later times some of our Roman generals, to say nothing more, have lost their armies by unskillfully neglecting this precaution. Two armies, one tired and spent, the other fresh and in full vigor, are by no means an equal match.

Again, emphasis here is on keeping troops rested and energetic, anything that could be seen as a warmup by modern standards could be interpreted as "neglecting this precaution" in the words of Vegetius.

The other early source, Strategikon, predates the period we're discussing but I think it's views are still very relevant to how Iron Age armies fought. It is also a useful source for it's detail, including (literally) "points which the general must consider before the day of battle" and "points to be observed on the day of battle". There is a lot of interesting information here, Maurice goes through the process of inspiring the troops, organizing squads, delivering orders, gathering intelligence, maintaining the army, but no where is the physical process of preparing the soldiers bodies for exertion mentioned. There is great care to ensure the army and horses are properly provisioned and watered but as for training or warmups leading up to the battle? Nothing.

Interestingly, Maurice specifically describes how the general must take care to not wear themselves out:

On the actual day of battle the general should not take on too many tasks. He might exert himself too much, become worn out, and overlook some really essential matters. He should not look downcast or worried but should ride jauntily along the lines and encourage all the troops.

Later on he echoes a similar sentiment for the army as a whole:

On the day of battle make sure not to march the infantry battle line a long distance, no more than two miles away from the camp, so they do not become worn out by the weight of their armor. If the enemy delays the battle, have the men sit down and rest until they are about to get close. If the weather is hot, let them take off their helmets and get some air. On such occasion they must not carry wine with them, for it will only make them warm and dull their minds. Water, however, should be carried in the wagons and given to each individual who needs it as they remain in formation.

To wrap this up, in a sentence: I can't find any evidence the medieval or earlier armies warmed up before battle. In fact quite the opposite, there was intense care taken to ensure men and horses were as fresh as possible when the fighting started. That being said, just getting from camp, to the battle line, to combat, could take hours (even all day) and cover several miles. In this case the men would be warmed up most likely, but this wasn't something the commanders of the time specifically thought about.

Sources:

Praecepta militaria Nikephoros II Phokas

Taktika Nikephoros Ouranos

De re militari Vegetius

Strategikon Maurice Tiberius

edit for formatting

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