Litter Training for Kittens

         Has begun and has been successful before leaving.
           Often there is no problem at his new home, but....

Your kitten will be going through a lot of changes. The kitten is not yet fully mature in his digestive system. The stress of the move   could make him have a loose stool.
Always continuing with the food that kittens system is accustomed to, this problem should go away by it's self in a few days, if not, consult your vet.
Using the recommended litter is important, a change can throw him off in figuring out what it is for. NEVER use clumping litter for kittens ( especially NOT for pregnant cats close to delivery time). If it sticks to their feet they will lick it off and could get an impacted stomach. Clumping is Fine for mature cats. I prefer a hooded litter box.
Your Home is new, your family is new. I recommend at least 2  litter boxes on each level of your home. While he will be exploring the whole house, he is a baby, and when the urge hits, he needs to find a litter box in a hurry.
Make sure kitten is kept in a small room when no one is home, put food, water and toys in there along with a bed and litter box. The less chances there are for accidents, the faster he will learn to continue good litter box habits.
For the first few nights keep the kitten in a smaller room such as laundry or bathroom with a litter box close by in the room where he sleeps,  If an accident occurs, clean it and sanitize with something safe to eliminate odor. Hydrogen Peroxide works well.
If accident happened, figure out why!  Stress of to much action? Jealousy over or feeling threatened by another pet? Being left alone? Pay attention and, when not supervised, put kitty in his special room to prevent this from happening again.

If a problem Arises

First. You MUST confine the kittens to one room, with no carpet. If they protest, you must bear with it. Maybe put them in a crate, or soft cat carrying bag so they can be beside you in another room, when possible hold them, or use a harness and hold them. Do not let them on the floor except in their little room. You MUST be diligent  with this. It is a bad habit that must be broken. If they have gotten away with this for a long time, it may take some time, a month maybe more to brake the habit.
Second. During this time of behavior modification,  you MUST get rid of all odor in the carpets . What is under your carpet, lots of padding that has absorbed the odor?? If so, it can be a worse problem.  The urine may have soaked through into the wood under flooring. I hope it is not that bad.... I find hydrogen peroxide works well, test in an out of the way area... if it does not harm your carpet do another test, after pouring the peroxide sprinkle with baking soda and scrub with a broom. It will foam like crazy, lifting out the stain and odor. after foaming, use a wet vac or towels to soak up most of the moisture. Repeat as needed. There are products with enzymes that are good also. ALL the odor must be cleaned up before they are allowed back on the floor. You can buy a small light that will detect any remaining odor or stain that are not visible tot he naked eye.
Third. Figure out what started the behavior problem. A member of the home leaving or being added? A difference in work schedules, new furniture, a new pet, changes in their diet or litter or litter boxes. If you have had them tested and they are healthy, especially a urine test, then dig for stresses.... Often this is a dominant behavior, marking their territory. There are many things that can cause psychological upsets in animals,  just like in humans.
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If you do this after only 2 or 3 accidents, it will  be much easier.