I'm quite biased towards the link I gave regarding Sivir's SS, as I know they update it frequently, and test and retest rather strictly to ensure different conditions are tested.
According to that link I gave, Brand's passive is not applied, triggered or refreshed if a spell is blocked. So if Brand hits with Pillar of Flame (W), you will take the damage and the passive. Then, if you block Sear (Q), it does not do damage or stun (effect if target is ablaze--and there is no reapplication of blaze). This is also clear in the information for Conflagration: Both casting on you and the AoE are blocked. If you are ablaze and get hit by the spell it will not trigger AoE.
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