Dealing with an emergency is stressful no matter what. Making sure that you and your loved ones can move safely out of the way is crucial. At some point, you may need to remove a cat from a dangerous situation immediately and safely. Here are a few safe ways to restrain a cat in the case of an emergency.
Scruffing is a way to subdue your cat while moving them out of harm's way. You have most likely seen a mother cat carry her young by the scruff of their neck (the skin on the back of their neck). When a mother cat does this it triggers a calming effect in most cats that make them docile and easier to handle and carry. By firmly planting the hand into the skin of the back of a cat's neck, it can have the same effect. (Scruffing is also an effective way to give a cat medicine.)
Do not carry a full-grown cat by the scruff of their neck, their neck is not strong enough to hold their body weight and carrying your cat this way can seriously hurt them. However, if you support the cats body while also scruffing them (pinching or squeezing the back of your cats neck) this can calm your cat and make transporting them out of harm's way easier in most cases.
The taco hold
The Taco Hold (coined by the ASPCA) is an effective way to carry and restrain your cat and protect yourself from scratches and bites. If you have a cat bed, fold the sides around the cat like a cape draped around their shoulders and then firmly hold the bed around the cat. This will help prevent the cat from kicking and scratching out of your grasp. Then you can use the scruffing technique to hold the back of their neck and subdue them.
If you do not have a cat bed handy but you have a towel or blanket, you can use that to restrain your cat. Ideally if you can lay the blanket flat, place the cat in the center of the towel and with the cat perpendicular to the longer sides of the towel or blanket. Pull one half of the blanket over the cat snuggly around their body leaving their head uncovered. Then fold over the other side, making a blanket cocoon or burrito for your cat to rest inside of. You can fold over the bottom half so that the cats rump is covered however you should never cover their head. Covering their head could potentially harm your cat by cutting off their airflow, in addition to frightening them. However, since their head is uncovered be aware that your cat can still bite, so scruffing can be helpful here too, or simply avoid putting your hands too close to their mouth.
If this is an emergency where the cat is injured or in immediate harm's way, you can toss the thick blanket or towel over the cat and quickly pick them up inside the towel to move it. This is, of course, more dangerous since a cat can still scratch and bite through the cloth with this method, and it is more disorienting for the cat. As soon as you have moved the cat to a safer contained area, you should remove the blanket from covering the cats face.
If your cat can move on their own and you just need to lead them toward a specific direction (like in the case of an earthquake or traveling outdoors), you can purchase a harness specifically made for cats. A collar and a leash can hurt the cats neck or even strangle them, it is also far easier for a cat to slip out of a collar. A harness, on the other hand, offers a firm grip that is harder to escape while being a lighter restraint for your cat.
Once you have moved your cat, they might be anxious. If this is the case you can use some natural sedatives to help your cat calm down. Otherwise, the best way to help your cat remain calm is by staying calm and talking to them in a gentle soothing voice. Finally, once they are safe and feeling better be sure to give them a snuggle—if they want one.