The chemical responsible for "the burn" is Capsaicin. The main reason you and everyone else has trouble with the lingering burning sensation is that it's not water soluble (which means it doesn't dissolve in water). So, flooding your mouth with water-based liquids doesn't loosen up those Capsaicin molecules on your tongue and throat.
Like @Iuls says, the most common and effective relief comes from full-fat milk or cream. That's because, while Capsaicin isn't able to be dissolved by water, it is fat soluble. It's also why more than a few people have been upset that the "milk solution" didn't work, when they tried it with skim milk.
If you're looking for a more "fun" solution, it's worth noting that Capsaicin is also alcohol soluble, which might explain all of that tequila and beer they sell at Mexican restaurants.
Well, that depends on the individual Thai dish or Indian dish and how it was cooked, of course. But I understand what you're talking about. However, the difference in heat sensations is not due to the kind of pepper employed. It's all about fat, really.
Frequently Thai dishes are made with fresh peppers, and have a lot of acid and salt in them (from citrus, tamarind, and other flavors) but very little fat (comparatively). Because of this, many Thai dishes have an instant burst of intense hotness which goes away realtively quickly. The paragon of this is probably Thai salads, like larb or green mango salad, which are highly acidic and very very spicy.
On the other hand, most Anglo-Indian food (familiar to Americans and British) is in the form of "curries" which use a slow-cooked dairy base (butter, milk, and/or cheese), and are spiced with dried ground or whole chile peppers. As a result, when you first taste them the fat conceals the capsicum from your tongue, gradually revealing it as your saliva breaks it down. Hence the "slow burn". (I've tried to find a medical reference for this to link, but have not been able to yet).
Similarly, Thai coconut milk curries can build up heat slowly and that heat sticks with you -- because of the hot peppers cooked in the fat of the coconut milk.
Incidentally, there isn't one kind of chile pepper used by either culture. The Thai have dozens of varieties of hot pepper and Indians have hundreds (as well as a dozen different regional cuisines, a few of which are not spicy at all). In the USA, these tend to get narrowed down to a handful of different pepper varieties (and substitutions like jalapenos) because of limited availability. The spice you call "red pepper" could be any of a half-dozen different ground dried peppers of varying hotness.
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I've lived (and eaten my way around) South East Asia for nearly the last fifteen years, and I think I know the effect that you're describing. Feel free to let me know if I'm off.
If what you're talking about is a slow and gentle burn that sort of creeps up, intensifies, and then mellows back down gradually (which is what I think you're talking about) then what you're experiencing is very likely an emulsion of a few types of peppers in a fat with a little acid, salt, and something sweet (sweet brewed soy, or even molasses).
A common street food is chicken gizzard over hot coals that's been marinated using the above method featuring bird's eye, serrano, calamansi (green, with zest) rendered fat (duck or pork) and something sweet. There are as many varieties of that as there are street vendors cooking up all of that goodness. I'm pretty certain it's the fat that causes the burn to linger longer on your palate.
Emulsification also helps maintain the pungency of chili in more basic (cream / curry) types of sauces in which it would otherwise mellow very quickly.
My experience is mostly in The Philippines, but I've been all over, and I don't suspect that it's done too differently in other places. This will definitely get you the sneaky kind of sauce you want, you just have to experiment with the types and ratios of chili. With sweetness and an acid, something like chipotle is going to introduce itself before habanero, and then take the stage again once the habanero has finished wrecking your mouth.
What remains is some experimentation that might ... err .. hurt a little :) But what you'll eventually end up with are a few really tasty pastes that you can add to any other sauce in order to get the desired effect. Also, don't feel bad about just using combinations of ready made chili sauces (sriracha, piri piri, etc) - this can save you a lot of work and give you a more refined taste. The sky and your budget for buying them are the limit.