Any new addition to your indoor or outdoor cage/aviary will need a period of quarantine in a separate, but nearby, cage. This gives you time to observe the new birds and make sure there are no health issues, and to let all the birds get used to each other. After the first week, if all is well, bring the cages close together so that the finches can come face to face.
The first physical contact without intervening bars should take place in the cage the birds are due to occupy together. If this is the older bird’s existing territory, swap the fixtures and fittings around to make it feel like a different, neutral space. Put food stations on both sides of the cage, to allow the birds to feed separately.
If there's any bickering or flapping around, don’t panic – this is natural, and as long as there's no persistent violence taking place, things should quickly calm down once the hierarchy has been established. Offer some of the birds’ favourite food to take their minds off confrontation via group feeding.
New birds will need to be caged separately for a while
If, after a day, the birds are fighting each other, you may have to keep them apart longer and try again in a week or so. The larger the cage, the easier the transition should be.