On occasion people will speak of some object "exiting one's future light cone". How is it possible to escape a light cone without traveling in a spacelike direction?

Usually people do include traveling in a spacelike direction as a component of the 'exiting'. But the alternative is for the objects ('you' and 'the other thing') to be at rest relative to each other but a long distance apart - while living in a universe that does stuff like this.

ie. Imagine ants on the outside of a balloon that is getting blown up at an accelerating rate.