A lot of what you should pack with you is based on what you know about your environment. It is not a matter of what technology you have available, but what is your level of awareness about disease, chemistry, human psychology etc. I am assuming a modern understanding of the environment, with iron age technology.
- Snakebite treatment kit. Includes herbs and potions.
- Small tweezers for pulling out thorns, spikes etc and a sharp scalpel for cleaning festering wounds.
- Bandages for wrapping on wounds.
Each person ought to carry their own medical kit. It would easily fit in a medium sized buckskin pouch, that can be hung from the waist.
The weapons would be used both for hunting and for defense against hostiles.
4 knives. Adults carry large knives (cutlasses), one made of obsidian, for skinning game and other bronze for chopping dry twigs for fire. Children carry one small knife (the size of kitchen knives we use these days). Both made of stone, primarily for defending against hostiles.
Two bows. One large, as primary long range weapon against hostiles and for hunting large game animals. Smaller bow for hunting smaller game (squirrels, hares and snakes). Bows are carried by the adults while quivers/arrows are carried by the children, considering the children are aged 8 or above.
Two blowpipes. 4 feet each. For hunting birds. Blowpipes would be hung by the children's back while their darts would be carried in the quivers with arrows.
A pouch of poison for applying on arrow and dart tips. The pouch would be carried by one of the adults.
While it is understood that you cannot carry all the food for your journey with you and that with your weapons you can easily hunt game ranging from deer to birds, you would want to make sure you don't starve in case you are unable to hunt anything for a few days.
Every time you are in a region with easy game (lots of ducks/turkeys/squirrels/hares/deer), you would want to hunt as many as you can carry, and smoke or sun-dry their meat for future consumption. While you might get lucky and bring down a bear with a well placed poison arrow shot to the neck or eye, I would strongly advise against it. There would be much greater chances of you getting killed in the effort instead, and you would not be able to carry all the meat with you, either.
Your dog is going to be a major help for you in tracking large animals you have successfully hit, and in retrieving birds you managed to shoot.
In places of shallow water, you would want to hunt small fish or water fowl. You would also want to forage for berries and edible roots and tubers when you find them. Seasonal fruits such as wild avocados, pineapples and apples would also be a big help.
In case you are able to exchange some valuable item (whichever you might be carrying for trade sake) for food, I would suggest getting corn or molasses in return. Molasses is essentially sugar and can be used in hard times. Corn is great for drying up and roasting in small quantities on nights when you don't find any game.
Considering that you have mentioned deserts in your journey, I would strongly advise carrying your water with you for desert parts. That would mean thoroughly washing a deer or goat's bladder, tightly tying up one end of it, filling it with water and then tying up the other end. Everyone would need to carry one such bladder with them for desert walks. Remember, you would also need to carry water for your dog, too.
Clothing And Shelter
4000 miles is not a small distance and on foot, with children, you would hardly be making more than 10 miles each day. This would mean no less than 400 days for the whole journey. And then there would be rain and snow, too. Of course you are going to need warm clothing and some supplies for building a shelter.
Your regular buckskin/wolfskin coats and loin clothes will do for most of the time. But in winter you are going to need something thicker and warmer. That is going to be either a full (head to ankles) length of a wolfskin coat, made from skins of two or more wolves. Another option would be bearskin coats. These would be warmer and heavier and would also provide protection during snow. However, as mentioned above, bears would be next to impossible to hunt with your 2-people team even with poison tipped arrows. This means you would have to get them from some village, in return of some valuables such as wild game or metallic objects (such as axe-blades).
Once the winter season is over, you can trade your bearskins back to some other village in return of supplies you need, so that you don't have to carry unnecessary warm clothing during summer time.
Sometimes you might need to stay in one place for a week or so. That would occur when there is a heavy rain lasting more than a day, resulting in boggy places and seasonal streams everywhere. You might also have to stop travelling in case one of your family members gets sick/injured and is unable to travel. In such times you would need to build a small hut or tent for your family. This would include chopping down wooden sticks and building a small hut with clay and sticks. It would take a day to raise such a hut if all of you work on it.
This is going to be the hardest of all. Since you don't know how to make fire from flints. You would have to carry a lit torch with you all the time. Amazingly enough, buying fire from a village might cost you valuables, even if it is burning in every hut. I would certainly advise carrying your fire with you.
In places where you find oil wells, make sure to dip your torch in it, as it would burn longer with the crude oil applied on it.