As with teaching and proofing any command you need to gradually build on the distance, distraction, and duration.
Stop, stay, sit are really all the same thing for my dogs. When given the "sit" command they are expected to hold that command until released. To proof a reliable sit-stay from a distance you first need to work on the command with the dog right next to you in low distraction environments. I like to keep the dog on a lead while walking at a heel and give the sit command by voice or through the whistle. Once the dog reliably sits on command while walking at a heel on lead, I move on to giving the command while dropping the leash and walking forward while the dog maintains the sit-stay. This eventually moves on to walking to a heel and sit-stays off the leash. We are still in a low distraction environment at this point.
Eventually I incorporate the premack principle, which is basically using what the dog wants to get the dog to do what you want (you can run off and explore AFTER you sit-stay). Using the premack principle is a wonderful way of teaching a dog that there is a time and a place for play or chase, but that you decide when is the time and where is the place. To proof a sit-stay using the premack principle I like to use flirt poles, dummies, or balls. I ask for sit-stays when throwing out something for a retrieve. Eventually I'll ask for a sit-stay after a dog has already been sent for the retrieve or right in the middle of a chasing round with the flirt pole. I like to use check cords at this point to still keep control of the dog just in case they run in (or off), which in the beginning stages they will do. At this point I'm still in a relatively lower distraction environment, meaning I'm in an environment where I directly control the distractions (e.g. the dummy) even though the distractions I'm offering may be high distractions for the dog.
Once a dog has proved to me that it has a reliable sit-stay in a lower distraction environment I will move to a park or a field and further proof this training by going back to the very beginning - heeling and sitting on lead. Eventually the premack principle is again used, but in this higher distraction environment (the smells, the squirrels, other dogs, etc). I like to use these distractions to my benefit and like to use a dog's specific drives to aid in my training. If a dog likes to chase squirrels and birds, I let them, but only after a good sit-stay and when I deem it is safe to do so. If a dog likes to run around sniffing everything that's fantastic, but only after he's done what I've asked him to.
Again, to be fair to the dog you can't expect such refined behaviors without first teaching and proofing them in a gradual way. Start slow and gradually move up to more challenging environments. If you can, use environmental reinforcers to your advantage by using the premack principle. It's so much easier to train a dog for these types of behaviors when you're working with the drives we've purposefully bred into them, instead of denying them. Consider getting a long line to help proofing behaviors from a distance and also consider whistle training your dog.
