Know how cap scoring works: Not all caps are worth the same, and it differs between 3 cap or 5 cap games. In most (~80%) of random lockdown games, this knowledge doesn't matter too much since weak cappers (underskilled or high ping players who die a lot) or non cappers (damage only players) skew the capping power of each team that it becomes onesided, but for close games knowing what to prioritize can mean the difference in winning or losing 501-497.

In 3 cap games: first cap is worth 5 points, second is 3 points, last cap is 1 point. Getting that last cap is not that essential for score control of the game. But at the very least it is worth neutralizing.

In 5 cap games: first cap is worth 5, second is 4, 3rd is 3, 2, then 1. Getting a 3rd or 4th cap is less important than neutralizing the enemy 1st or 2nd caps, but because these maps are typically large it is faster to complete caps than to hop around neutralizing caps (especially if they're on the other side of the map).

The first cap is the most important: The first cap (in either 3 cap or 5 cap games) is the most important - that first cap generates 5 points per tick and every subsequent cap generates less. Your high level objective is to always maintain having 1 cap in play and trying your best to neutralize (not necessarily cap) all of the enemy ones. In many cases it is worthwhile to secure at least 2 caps for 3 point games (3 in 5 point games) not because that extra cap gives you a lot of points, but so that you don't lose as much point growth if an enemy successfully neutralizes a cap.

Do your best to neutralize a cap during a fight - don't necessarily need to cap: Related to the above point, there are a handful of times when I've won due to decapping the last enemy cap when they're at 490 or lost because we couldn't decap it fast enough - either I got there too late or teammates are busy managing the fight off screen. The saddest is when I see teammates fighting around the cap when they could have decapped and stalled the enemy from winning.

Don't clump together in a cap: Anticipate being dive bombed by a GF double swing from below, and AT ping advantage will secure being hit even if you react fast enough. Even more applicable if you're aware of any recons on enemy team in play. Try to stand along the edges of the allowed capping range to maximize spacing out of you and your teammates along the cap.

Guardian: In the face of haze bombs providing area of denial, the shield is for stalling: Guardian shield ticks in haze bombs are worth pointing out because it allows guardians to sit practically invincibly on a cap. Assuming enemy haze bomber is also on the cap, that stops the cap from changing ownership - ie. stalling. Understand when it is appropriate to stall - assuming you have no backup to fend off the haze bomber, stalling actually helps enemy team when they own the cap. Since the cap won't change ownership, stalling just helps them earn points off that cap. Don't stall just because you can - sometimes I see enemy guardians do this and I leave the haze bomb/guardian stalemate alone because it actually helps my team. Stall when you have a team lead to prevent enemy team from gaining ground.

Underused: In team capping (ie. 2 or more people on a cap) in a cap that has a bottom screen blind spot, have one person scout the bottom of the screen to prevent ambushes. This is something I admit I don't do enough, but understand its value - on the one hand, that 1 person could have spent the time speeding up the cap by helping out. On the other, I believe it is more valuable to stay in bottom screen as "lookout" to prevent being ambushed from below with a GF combo or charge attack and risk having the cappers killed. Essentially the scout "protects" or helps anticipate enemies who would otherwise disrupt a cap from succeeding. Becomes incredibly useful when enemies are smart about using bottom screen advantage, and there are some players who are really good at securing mass kills by surprising cappers from below.

Stay on the damn point, get the enemies off it
^This. I've facepalmed so many times while spectating because people somehow do not understand this. Someone literally ran off a point while it was half a second from being captured to chase a guy that wasn't even threatening the point. Why do you even bother trying to cap if you aren't going to commit to it?

If you are the best player on the field and can kill everyone on the opposite team. It is by far the best strategy to do so, letting all your team mates cap in peace.
I would say this is only true on small maps and if your teammates have almost no chance against the enemy. The exception is if the enemy team is full of damage farmers.

I would like to add:
Adjust your aggressiveness to maximize your team's presence on the field
If you're the only one alive on your team, you want to push a point close to the enemy base. There's two scenarios that can occur:
a) They kill you, but that leaves them on their side of the field, buying your teammates time to come back.
b) They are gathered at your base, so you successfully take the point, bringing their attention back to their side.

If your whole team is alive, but most of your teammates are on low health, do NOT rush to help them. Instead, hang back until they die, then go in and take their place. It's tempting to have the whole team push the enemies into their base, but unless it's endgame (or horribly unbalanced), it usually ends with your entire team dying and a reversal of the situation.

Two is company, three is a crowd
Two people capture much faster than one. Three people is too much, mainly because it statistically doesn't leave a lot of teammates on the rest of the map, and it also doesn't give you that much more edge than two people in a skirmish. The exception is if you are traveling with a good guardian and capture systematically.

I've seen carries in 4v4, but it takes a special type of person to pull it off.

Carrying depends more on how the level of your opponents are, as everything does in LD to be fair. For example, I can carry a disadvantaged team in 4v4 even if they're bad so long as the enemy team is also quite bad. However, throw someone like Scirio on it and I'll have my ability matched and I'm not nearly as significant to my team because I'll probably end up with a lot of deaths to Scirio.