1) Some desirable ends are self-defeating to directly aim at. (This is plausibly true of both happiness and maximizing utility, for example. More obvious "essential byproducts" include spontaneity and sleep.) Because the ends are desirable, direct rationality tells us to aim at them. Indirect rationality suggests we should adopt a different, globally optimal, strategy.
2) Parfit's notion of "rational irrationality" can be made more clearly coherent by noting that it can be globally rational to induce local irrationality in oneself.
3) Applied to epistemic deliberation: local rationality tells us to consider the content alone, and assess each argument or appearance on its merits. Global rationality may invite us to take properties of the source (or 'vehicle') into account. For example, we may reject the evidence of our senses if we've just taken hallucinogenic drugs (or been possessed by the devil), or we might reject an argument on the basis of the speaker's known unreliability. More generally, considerations of 'meta-coherence' concern global rationality. Huemer writes:
I think there’s a rationality constraint roughly to the effect that one’s first-order beliefs (/degrees of belief) should cohere with one’s assessments of how good one’s belief-forming methods are — so if you believe P (to a high degree), you must think that your method of forming that belief is highly reliable (you must also think your belief is justified, undefeated, fully grounded, etc.).
There’s a threat of violating this condition if you assign a higher degree of belief to what seems right to you than to what seems right to person S, while you also hold beliefs that imply that S is more likely to form correct impressions about the matter in question.
Direct rationality leads one to accept "what seems right to you" on the basis of immediate evidence. But global rationality might recommend an indirect strategy of deferring to experts, if that would have globally optimal results for one's pursuit of truth.
4) Regarding Newcomb's problem: Local rationality endorses dominance reasoning at your present moment, and thus recommends taking both boxes. Global rationality again recommends the strategy which will bring you long-term riches, i.e. being a one-boxer.
5) Relatedly, the global rationalist can beat Kavka's toxin puzzle, since they will follow through on their intention to drink the mild toxin even after they've already received the reward.
There are a whole raft of other related problems for local rationality. Consider Quinn's self-torturer (a sort of forced-march sorites paradox), or Pollock's Ever Better wine:
The wine slowly improves with age... More good news: you are immortal. Consequently, you are indifferent as to when you consume a particular good. When should you drink the wine?
Not now, the wine will be better later.
Not later. For at any given time it will be true that the wine will be even better if you waited longer.
But if you do not drink the wine now and do not drink it later, then you will not drink it at all!
-- Roy Sorenson, 'Paradoxes of Rationality' (in The Oxford Handbook of Rationality), p.261.
In such cases, one can only succeed by making resolutions and sticking to them in spite of their local irrationality when the time comes. So that's what global rationality recommends.
Can you think of any other examples of this distinction in action?