so it would be less useful to the destination star.
Sure, but I don't see a problem here. If you have a particular attachment to a star system, you may want to supply it with energy from other stars instead of moving to them.
The point is that since the energy will get "consumed" elsewhere, the waste heat at the origin will be minimal and so the Dyson sphere will remain relatively cold (though higher than the background temperature, to be sure).
However, the destination star would emit large amounts of heat, i.e., infrared.
This thread is for asking any questions that might seem obvious, tangential, silly or what-have-you. Don't be shy, everyone has holes in their knowledge, though the fewer and the smaller we can make them, the better.
Please be respectful of other people's admitting ignorance and don't mock them for it, as they're doing a noble thing.
To any future monthly posters of SQ threads, please remember to add the "stupid_questions" tag.