Escapes
Update #6, released August 22, 2013, added Escapes to almost every mission. After accomplishing your objectives and getting to the getaway van, you'll be taken to the post-mission screen as normal. However, rather than getting your payday and final XP, you'll only get that day's XP, and no cash. Then, after readying up, you'll be taken to the escape, which is the second part.
At this point, you'll have a chance to change your loadout. This is in your best interest, because all items and gadgets reset, and you have a chance to rearm. Because the cops are on you immediately and stealth isn't an option, your best bet is to just grab your best Weapons and practical deployable Equipment. For example, if you had selected the Two-piece suit and a weak, concealable primary weapon, now is your chance to change to a Heavy Ballistic Vest and an AK Rifle. Plus, anyone who brought an ECM Jammer to the main mission can now switch to an Ammo Bag, Doctor Bag, or Sentry Gun for maximum efficiency.
Escapes can take place on one of several maps, and at a couple starting locations in each. Your getaway van is overturned or at least run off the road, and you'll find yourself near it with your crew. If there were any large loot bags, they'll be scattered near the van. Be careful around the van; it's often on fire, which will damage your health and can kill you.
There will be an initial deployment of cops around your immediate area. Take them out, then pick a direction and start walking with your loot bags, if any. Cops will deploy, often in assaults, but they tend to deploy around your starting area. If you can get across the map early, they may deploy so far away from you that you can pick them off before they get too close. The cops' strength will depend on the difficulty level of the chosen mission; one-pip missions will likely see no special enemies, while three- or four-pip missions may see Tasers, and higher ones will see Bulldozers.
After about four minutes, a second getaway van or a helicopter will arrive. Toss in your loot bags as normal, then stand in the white zone to leave. You'll have a short window when the getaway van will simply leave, even if you're not near it. The timer scales depending on how many bags of loot you have, giving you a chance to bring it all. For example, if you have no bags of loot, the van often leaves after 60 seconds. Each bag of loot tends to add 30 to 60 additional seconds. If you're playing solo, the time window is usually large enough that you can get all the bags over there, even if you have to make multiple trips. However, as always, the longer you wait, the harder it will be to escape.
Note that you will not have to bring in a minimum amount of loot, even if the original mission required it. For example, in a standard-difficulty instance of Jewelry Store, you're required to bring three bags of loot. If you trigger the escape, then you will not be required to secure the three bags, or even any bags. However, the payout will downscale accordingly, so if you bring no loot at all, you'll be paid almost nothing. Note that any instant cash items, such as paper money or jewlery, will count regardless.
Although most missions have escape sequences, some don't, and some are avoidable. Often, if the cops are onto you when you get to the getaway van in the initial mission, especially if a police assault is active, you're guaranteed to have to deal with an escape. Conversely, if your escape vehicle is a helicopter, such as in a long Mallcrasher instance, you will not have to deal with an escape. Also, for missions where you can stealth through it such as Framing Frame - Day 1, or missions you get through very quickly such as a perfect run in the Ukrainian Job, you will not trigger an escape sequence.
As of Update #11 (September 3, 2013), you can buy an "Expert Driver" asset for almost every mission that will prevent any Escape sequence from taking place. The asset will only appear if you have the Inside Man skill from the Mastermind skill tree.