Well, I have no problem with your last paragraph, but I think you overestimate the mobility of medieval armies. "Finding local supply" basically means sacking

Well, I have no problem with your last paragraph, but I think you overestimate the mobility of medieval armies. "Finding local supply" basically means sacking towns, which is not always a viable option (ask Napoleon), especially in a war that lasted seven years. The Japanese army did succeed in initially overwhelming Korean forces with supreme military tactics, so much so that they entered Seoul basically unscathed, only to find that the king and his courtiers have fled north. From what little I know, it seems that the Japanese forces didn't quite understand the situation. In traditional Japan (from what I've heard), if your force wins the battle and enters the enemy castle, you have won. The enemy's people are now under your rule and their food and supply is yours. Instead, Japanese forces found itself surrounded in a sea of hostile troops and militia bands, who will (incredibly) fight to the bitter end. Not unlike what the US found in Iraq, but without the ridiculous military advantage. Korean army's major military victories included the battle of Haengju (near Seoul) and Jinju (southeastern Korea, near present-day Busan), which all happened within one year since the war broke. A major reason why Toyotomi could parley for truce that involved Japan securing the southern part of Korea, was that Chinese forces had no reason to stake their lives for someone else's country, yet they still had enough political power to force Korea to the talking table.