There are a few problems with viewing things like Battle of the Nations (and SCA and a variety of other similar activities) as historical combat. They're historically themed, but ultimately are modern combat sports, not historical recreations of battles. Armour and padding tends to be extremely overbuilt, is often only vaguely historical, and nobody is actually trying to kill you, they're trying to beat you. It's better to think of BotN as more like tournaments of the period, rather than actual martial combat. Such activities have a tendency to whale on each other with blunt weapons, until someone "loses," rather than trying to recreate historical combat.

More specifically to the question, to focus on approximately the mid-late 15th centuries (based on your comments on weapons and armour), there are a few ways to kill or incapacitate a fully armoured opponent.

N.B. that I'm speaking to martial or "real" combat, not tournaments: if tourneys are what you're asking about, I would defer to someone with more specific knowledge on the subject.

1. Get them off their horse, if they're on one. That's the big first step. I know you specifically stated hand-to-hand foot combat, but getting to that point is vital.

2. Flexibility and movement is extremely important when your life is on the line. That's why, in field armours, you'll usually see plate on rigid parts of the body and maille in places like the groin, behind the knees, in the armpits and elbows, and around the neck. You'll see extremely intricate flexible plate defenses in armours such as those in the slightly later Greenwich style, for example, but in the majority of field armours you'll see concessions for mobility.

3. Getting into those gaps. Half-swording is a longsword technique that refers to, essentially, grasping the sword halfway down the blade and the handle and using it as a lever and fulcrum, using it to direct the point of the blade in a very pointed way, essentially turning your longsword into a very large, very pointy dagger, that can be used to pierce through the vulnerable maille areas. Similarly, martial daggers of the period, such as late rondel and bollock daggers, were also more pointed than blade-sharp. Period riveted maille was strong, but if your opponent had been thrown off their horse and was laying on the ground, could be pierced with enough effort.

4. If you can't get into the vulnerable flexible areas, you can target the plate itself. Plate armour was effective, but so were the weapons designed to beat it. Pole-arms of the period, such as polaxes, maces, and warhammers, were capable of crushing plate to such and extent as to render the person inside incapable of fighting. If you look at period pieces you'll see a lot of pointed surfaces that can concentrate force and crush the armour, which depending on force or placement may actually damage the wearer, or deform the armour to inhibit movement, leading you back to number 3, someone at your mercy with vulnerable points.

This is in response to contextualize your impression of how Battle of the Nations - I'm going to loop the Society of Creative Anachronism in here too, since I have more experience with that, and talk about both - combat actually works, because I think that may help your understanding of what those activities are trying to recreate.

First, to start with, as helpfully noted by others neither activity is meant to fully recreate period combat. The authenticity rules are relatively relaxed, in the Battle of Nations it is "represent a complete set of one time period within a span of no greater than 50 years within a region." SCA is everything needing to be prior to 17th-century, and generally the rule of thumb is to appear 'within period' by someone standing 20 feet away.

Both organizations have guidelines that are designed to make the combat appear and feel as accurate as possible, but safe. For example, neither organization allows thrust weapons, no matter their historical accuracy, as they can cause serious harm even if blunted and through armor. I will add here that the BotN is a bit less strict with safety than many other historical fighting organizations, such as the SCA.

they are swinging full force, yet nobody ever gets seriously injured

Well... sort of. In BotN, when wearing heavy armor like full plate or chainmail, they do swing with much harder blows than in the SCA or when wearing leather or cloth armor. I personally know several fighters who prefer and relish in their experiences in the BotN because of this, and to compensate BotN has stricter armor guidelines than the SCA does. Those stricter armor guidelines tend to be pretty cost-prohibitive for most people, which is a big reason they aren't used everywhere. In the SCA and other more narrow historical fighting organizations, and to some extent still present in BotN, I want to be clear - those blows are measured. It may not look like it from the outside, because it is still exhausting and happens at full speed, but there is a definite range for an acceptable blow. The blow has to be strong enough to be noticeable, where it may be plausible if the weapon was sharp it would have injured, but definitely not full-strength because even with all the safety guidelines in place that could still seriously hurt someone. And, although rare, injuries still happen. Usually because people are not following the rules, think they are too smart for the rules, or... because they aren't calibrating their blows. Even a blunt weapon can bruise some ribs swung hard enough, and that is not the goal here.

In both organizations for each type of combat there are different types of blows which are considered 'killing' blows. This depends on multiple different factors. What kind of weapon is it? Some weapons have larger "killing" blow ranges. For example, a mace has pretty much any contact between the knees and the neck. (Hitting below the knees or the head is usually not allowed for safety reasons.) This is because those types of weapons, swung at actual full strength, will crush through bones even under plate armor. In medieval times, a crushed hip, damaged organs, and broken ribs meant you were going to die. At the very least, you would be so incapacitated that you could not continue the fight. An edged weapon, such as a sword or axe, has different killing regions - usually just the torso, and it must be 'on edge' when hit and to a degree of force considered 'stout' or 'noticeable,' depending on the org and who you ask. This is mostly because although an edged weapon wouldn't get through full plate at any old spot on your armor, having more specific guidelines would become impossible to police.

Most of the areas that you would actually be aiming for to kill someone in full plate are understandably off-limits for safety reasons: face, back of the neck, groin, back of the knees, armpits, etc. Removing an opponent's armor, twisting a limb, actual hand-to-hand combat or wrestling, anything other than using designated weapons is not allowed.

Armor was pretty effective, but not as effective as the BotN would have you believe.

As a previous member of the SCA, if you live in the Known World and have any interested I highly recommend you check it out! Even if you don't want to do fighting such as they have on BotN, there is rapier which is less intense (what I used to do), but also lots of fun cultural activities. They are as a group incredibly enthusiastic about their craft, and if you approach them with a genuine interest in becoming involved chances are they will do a lot to help you participate and feel welcome.

I'm going to pull from Fiore de'i Liberi for this. https://wiktenauer.com/wiki/Fiore_de%27i_Liberi#Sword_in_Armor

Basically if you want to kill a knight in armor with a sword you had to exploit a natural weak point in the armor. Fiore preferred going through the visor or under the chin. Of course that was something that an opponent wouldn't like, so there were many published and trained techniques to take an armored opponent to the ground using joint locks similar to modern jujitsu.

Halfsword techniques where an armored combatant grips his own sword by both handle and blade can also be seen. These techniques attempt to pierce the armor at a joint, generally either the armpit or groin. The two handed approach allowed a combatant to control the point and put full body weight into the attack once the outer plate was defeated.

https://wiktenauer.com/wiki/Fiore_de%27i_Liberi#Axe_in_Armor

The axe here refers to a poleaxe. It still needs to exploit weaknesses in head armor when a knight is standing. The poleaxe does have advantages of reach and leverage but still isn't a truly effective killer against plate.

Once a knight is down then a heavy weapon like a mace can hammer them against the ground and eventually cause death by crushing either the head or chest. You'll note that in Battle of Nations they stop the fight once a combatant loses their footing, that's because the lethality of the combat increases once person in full armor harness is down.