Five months, 6 weeks, and 4 weeks. You are in Florida so the weather is warm. Cold is not a concern but rain might be for the 4 week olds. And thanks for the size of your facilities, that saves me asking.
Are your 6 and 4 week old getting along? If they are I'd treat them as a unit so you don't have to integrate more later.
Until they are all mature, older chickens outrank younger chickens in the pecking order. Sometimes they can mingle quite well but often the younger get pecked or worse if they invade their personal space. When that happens the younger chickens usually quickly learn to avoid the older. They do not approach the older and if the older walk toward them they run away. What did it look like when the older chased and pecked the younger? Were they fairly close when it started and did it stop when they ran away? That would be normal.
If you read the posts on here you will soon realize we get all kinds of different results when we try to integrate chicks, whether with older adults or just older chicks. Sometimes they start to mingle immediately. No problems at all. They may eat or drink from the same dishes or even sleep together. This isn't very common but it happens. Sometimes they reach this stage in a few weeks. I see several stories on here where this happens.
What I normally see with my flock, facilities, and management techniques is that the juveniles avoid the adults until the pullets start to lay, then they become one happy combined flock. I have over 2,000 square feet outside but my coop is smaller than yours. During the day the juveniles avoid the adults like the plague. At night they all sleep in the same coop but the main roosts are for the adults only. The juveniles find some other place to sleep away from the adults.
Your facilities are huge for 8 chickens compared to many on this forum. You are going to make this work. But imagine you are your 3 older chicks (spread out a little) and stand in the middle or your run. How far away can a group of 5 younger chicks get and still be in the run? It is not real far.
So what can you do? Give them as much room as you can. Try not to confine them in a small space together. With your facilities that should not be a problem. Improve the quality of what space you have with "clutter". Here, clutter means stuff they can hide underneath or behind. It means perches where they can fly up to get out of the reach of the older. This might be an old antenna dish or plywood up on cinder blocks they can get under. Maybe a piece of plywood leaning against the fence open on both ends and attached so the wind doesn't blow it over. Or some kind of perch they can fly up on.
Have multiple food and water stations, separated and preferably out of sight of each other. That way the younger can eat and drink without being bullied by the older. I have one food and one water station in the coop and a few scattered outside.
I don't know what your small coop looks like. If you can, move it into the big coop where it is predator safe and leave it in there for a few days. Have the younger chicks sleep in it at night so they learn that is where they are supposed to sleep. Then one day when you can observe, open that coop and let them out. The way I do it is to let the older out first so they can get to the run. Give them as much room as possible. And then observe.
Good luck!
