Sounds interesting. How would Let it Ride apply to a challenge that takes a long time, such as climbing a mountain.
It is difficult to discuss without stakes and framing, and thus the consequences of failure.
First, it entirely depends on how big a scene you want it to be--or maybe even multiple scenes, if the mountain is the backdrop to other action. If this is a bit of a challenge before you get to the summit and start the next part of the adventure, it might be one roll (or in D&D, a set of rolls). Or you might have trudging on the lower slopes one thing, while navigating the snow near the peak is another thing. Let it Ride applies to the scene as framed, which means how you frame it determines how long the roll "rides".
(In Burning Wheel, you may also allow a roll to stand even when circumstances change, but the mod may change the outcome. You had enough success to climb the first part. Then the fight with the snow apes on the slopes damaged your equipment. You don't reroll the climb check, but your result is modified to show that your equipment is damaged. This is largely a matter of taste in d20 applications, since unlike BW, they don't have major resources embedded into the original roll. The closest in 4E would be someone using a daily power or item.)
Second, the way I handle something like climbing is to make modifiers to the roll(s), based on how long the climb is. Then the consequences of failure are agreed based on what those modifiers are.
For a simple example, there is a 150 foot, near sheer climb, with no good places to rest better than a narrow ledge with a piton hammered in. Failure is falling, the lower the overall roll, the higher up when the fall occurs. We might start with a +10 to the DC for this. Then the party deploys their ropes, climbing materials, magic, and so forth. There is no pressure in this example except the risk. So they take their time (+2 circumstance mod). But it's a long climb, so a group Endurance check is required (+2 or -2 depending on results). They probably choose to use a certain amount of their resources to mitigate failure (e.g. tie that poor wimpy wizard off with ropes). I'm probably leaving stuff out, but you get the idea.
It's all negotiated based on the fictional challenge and the resources the party brings to bear. Then whatever rolls are in that are rolled, including one climb check per character, to establish the baseline. You either make the climb without injury, you get hurt but the party gets you to the top anyway, or if everything really goes sour, someone might die. The Let it Ride part happens in that you don't call for the climb check multiple times.
For a more complex example, consider a 50' tower of rough stone, climbed at night, with guards on the walls. Failure is getting spotted (with predictable consequences). Sucess is getting over the wall without being spotted. Certainly, the climb check factors into success, as making an easy, quick climb minimizes exposure. But the main thing is the sneaky party. Unless someone rolls a 1 on their climb check, a failed check means that the person slides several feet (before someone's rope saves them from a fall), drops a piece of equipment, etc.
Now, to make the "Let it Ride" part more interesting, let's say that the tower is an outer wall, but the party needs to also climb down the wall on the inside, sneak across the court yard, and then climb a shorter wall into a third floor manor window (this being their plan). Here, there is still only one Climb check per character (though obviously other checks for being sneaky, observing the guard patrol, etc.) Make a good climb roll, you do the climbing part alright. Make it really well, you do it quickly, thus making it easier to be stealthy. Fail it, the other stuff gets harder, but you still get over the walls.