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Alejandro1 comments on Stupid Questions Open Thread Round 2 - Less Wrong
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Suppose (just to fix ideas) that you are at rest, in some coordinate system. Call FLC(t) your future light cone from your space-time position at time t.
An object that is with you at t=0 cannot exit FLC(0), no matter how it moves from there on. But it can accelerate in a way that its trajectory is entirely outside FLC(T) from some T>0. Then it makes sense to say that the object has exited your future light cone: nothing you do after time T can affect it.