Moving Kitchen sink to opposite side of room
There isn't only one proper way. If it's convenient to do so, extend the vent straight upward. If there is attic and roof immediately above then you could make a new roof penetration for the vent and detail with a pipe jack flashing in the usual way. If there's another story of the building above, you could get into that ceiling/floor space and take the vent anywhere - continue upward, route to meet with the existing vent, etc.

If taking the vent up is problematic, route it downward instead. It could be arranged as for an island sink as explained at Family Handyman (they mention two methods: loop as per the image below, or use an air admittance valve).