Here would be my steps, based on successful cases of continuous improvement I've seen implemented in level 5 CMMI shops. I believe processes like Lean Six Sigma focus on a similar approach, although (based on casual observation) with a slightly different nuance.
Examine your current state
I bet that it's a no-brainer in the cases that you mention that harm is caused when people don't do the process properly. The big question is how much harm? How often does this happen? What is the magnitude of impact when it does happen? How does that impact affect the big bottom line? Everyone probably understands that these problems cause "some impact", but when you can put some hard numbers to it, you can see how big a problem you've got. Saying "we loose customer faith when we don't update change tickets" is one thing. Saying "we could have sold X many dollars more of our product/service" if customers had more faith in our process is a lot more urgent. It also tells you how much money/time you can spend trying fix the problem. No one wants to spend $100 worth of effort on a $20 problem.
Also - however - look for patterns in your failures and successes. I'm willing to bet that not EVERYONE does this wrong. Or, if everyone does it wrong, do they only do it in some circumstances? If you have problems with a specific group, or under specific conditions, see if there's something you can do to address that specific case, rather than trying to change situations that don't need changing.
Involve the People Who Need to Change
When you've got the data, get key people together - you may end up with a few meetings, if you have a big organization. One meeting is the people who need to change, who need the option to discuss this in a penalty-free way. In other words, they need the latitude to grime, complain and come up with wacky ideas without management freaking out at them. But they probably need a moderator to channel the discussion to productive ends instead of just complaining.
You may also end up with meeting with other key people - if it's the tool that it causing grief, you may meet with those who maintain it. If it's a two-group problem, you may end up meeting with both groups separately and then together. The big deal is to be cognizant of the fact that different groups come to different conclusions depending on who is in the groups. Use that to your advantage, not your disadvantage - the wrong people put together too early in the process will kill it flat out, but the right people will come up with some brilliant ideas that may even be cheap and easy to implement.
More important is giving people the chance to feel like they took a shot at solving it - most successful programs for change involve the people who need to change so that the effort becomes their effort and not something foisted on them from above.
Monitor and Give Feedback
Whatever metrics you used in the first step are ideal here - which means that although you may be monitoring the low level (# of status updates, etc) - what you REALLY want to pay attention to is that bottom line, as well. You may make the detailed change you want and still not get the payoff you are looking for - and it's important to know that. In my mind, this connection is what separates good managers from the pointy haired boss in Dilbert - that they are clever enough to look for the impact their people make and point it out to them. There is nothing headier than a victory in this area, and knowing you have failed helps you to figure out what you can discard as useless the next time you want to see change happen.
When giving feedback, consider that how you give feedback and to whom has a huge impact on the effect. Saying "this is urgent" when you body language says "I don't care" can be worse than saying nothing, and mailing the team when only one person is failing to comply reassures the one person that this is a group-wide problem when it really isn't.
Carrot or Stick
You'll see almost none of the work involves either an incentive (carrot) or punishment (stick). The modern management take on this is that how you lay out work, and how you get people involved in it is more motivating and effective than the carrot and/or stick approach. The general thought is that most people want to do their jobs, they just need guidance on what "doing the job right" means.
There's an exception for every case though, as you get into this approach, you'll see it takes a heck of a lot of time to get it done right... it's not a 20 minute task by any means, and some of this (like the metrics collection) is intense, data-driven work that most managers don't have time to complete on their own... so you're going to need help. In addition, you're probably going to hit the outliers on your bell curve, who really can't manage to do the thing you want them to do, even when the rest of your demographic has adequately turned the corner on making change a reality. That's where the carrot/stick strategy may work out:
Carrot - when someone goes above and beyond the call of duty. It is not above and beyond if you say "do this" and they do it adequately within their skill set. Incentivize "doing the job" and you'll be in a never ending, demotivating cycle. But if someone suggests and then seamlessly implements great new ideas, taking ownership and going above and beyond to nail down issues above their pay grade - make sure this gets rewarded come bonus time. And be specific in your praise, since you want to make sure that this excellence continues.
Stick - when you have ended up with an outlier that simply can't do the work that everyone else can do, it's time for the stick. To be effective, a stick has to be a real penalty. Doing the lame work that no one likes is not really the penalty you are looking for - it can easily be misinterpreted, and you can end up with an employee who never really does the job correctly. You're going to end up in the realm of docked pay, negative performance reviews and (if the issue is big enough) talk of termination. The things you mention are part of doing a good job, they are not "bonus work". They are having a negative impact on the business, and if this guy can't do what other people can do, chances are you can find someone else who can perform better on the open market.
I'm harsh on both the carrot and the stick here, because the two issues mentioned in the question are baseline for a good process. Both of these can have serious impacts for the company and it's in the mainstream expectation for the average worker to be able to both of these properly. If you were talking about a new "ideal practice" that was innovative and therefore had a higher chance of not paying off, my thoughts on what the bare minimum that the average person can do right off the bat may be more lenient. That said, when you've innovated and proved it's a huge value-add and you have people who can't come up to the new par... do you really want to pay them for sub-par work?