It's always a good idea to have a first aid kit at home so you'll be able to bandage any scrapes or bumps while preventing infection. Once your international movers have helped you get settled in, you may want to think about compiling all of these items so you can make the ideal first aid kit.

Anti-infection Solutions
Antiseptic ointment is necessary for preventing infection. Before you place a bandage on your cut, add some ointment to kill any germs. If you skin your knee, you can help eliminate the chance of infection by softly rubbing your wound with some alcohol wipes.
Bandages
In order to help a cut heal you need to protect it with bandages. Include a variety of different-sized Band-aids in your first aid kit (waterproof ones are the best) and get some gauze with adhesive medical tape for larger gashes. Having a small pair of scissors is also helpful for cutting your bandages so they'll be the right size.
Instant Cold Pack
If you get a bump or bruise, you can help reduce the swelling and numb the pain with an instant cold pack. These can be found at most outdoor gear retailers.
Sanitation Gear
To prevent the spread of germs, your first aid kit should also include sanitation gear. Medical-grade gloves can help protect your hands if you're helping someone else, and before working on your own cut, you may want to put some hand sanitizer on your hands.