Did the 2 armies just play a game of cat and mouse until one caught the other?

This is exactly what happened. Remember, armies don't just march around at random. They are moving towards a particular objective (even if that objective is simply "raid territory held by the enemy") and are limited in their options for travel due to geography. Thousands of men, horses, and wagons cannot easily pass through a heavily forested area unless sufficient roads are cleared. Rivers need to be crossed at particular points unless you want your horses to drown halfway across a ford. Armies are also limited in where they can travel by the constraints of supplies: men need to have food and horses need fodder. So a medieval army has a limited number of routes it is likely to take. Scouts and outriders would be deployed from both armies in order to capture prisoners, interrogate locals, and observe the enemy. Medieval armies would be surrounded by a cloud of horsemen going back and forth. Some were scouring the area for supplies and for forage, while others were directly seeking out the enemy in order to determine their intentions. Skirmishes between these parties of outriders and scouts were common and they would increase in size and scale as the two armies drew closer. Prisoners captured during these fights could potentially be valuable assets in determining the enemy's intentions.

How the beginning of a battle developed depended on the nature of the preceding campaign. Sometimes both armies arrived at the same location at about the same time. On other occasions, one army was rushing to catch up to the other, which placed them at a disadvantage because the defensive side might take the time to pick their own ground and prepare defenses. Medieval armies would often wait to fully engage, sometimes taking an entire night to rest and prepare their troops. Being on the defensive could be extremely advantageous, especially given the difficulties medieval armies faced with discipline and command, which meant that commanders were generally not over-enthusiastic about engaging immediately. Commanders wanted to wait in order to properly organize and prepare their troops for a full-on engagement. This waiting game could have deadly consequences. The French had this exact problem at Agincourt when the English took advantage of a lull before the battle to move their positions forward. Caught unprepared by the sudden advance, the French cavalry was unable to properly execute a flanking attack as they had planned and found themselves outgunned and undermanned when they tried to carry out their charge.