Most dogs do just fine with the proper equipment to assist in lifting and moving. Particularly with larger dogs, the biggest problem seems to be what to do when they become immobile.
There are several options. A handicapped dog can live a quite full and fulfilling life with proper support! Obtaining slings and lifts, or even a wheelchair to assist with mobility can give you several more years with a beloved pet. One of the happiest times in Hoover's life was when we took him (along with several other dogs) on "vacation" up north to scout property and go to a dog show. At the time his rear was quite weak and he required assistance to walk. I wish I had video of him attempting to run in the sand dunes with me trying to keep up holding up his butt! I bet I looked pretty silly, but he enjoyed himself completely and all it took was a simple sling.
A very self-sufficient dog may at first resist your help - but once they realize they have greater mobility (and more fun) with a cart, sling and assistance, most will go along with the program. Many dogs will make great effort to "help" when lifted and feel embarrassed when they have accidents. Many of these products make it a lot easier to care for older dogs that slowly lose the use of rear ends, became incontinent, or suffer injuries or joint problems. Vets rarely tell you that orthotic appliances are available to dogs that lose limbs or need support (even temporarily) for slipped patellas, hips, and hock joints. With temporary support they can often regain full use of the affected joint because it can heal without additional injury.
Orthotics allow your dog that might normally be on required "crate rest" to do mild exercise without further injuring himself. There are many kinds of orthotics that allows joints to bend and strengthen while healing. They help avoid loss of muscle mass during recovery and are lined with a soft padding to provide skin protection from irritation or abrasions while doing many of his normal activities. Other kinds of braces allow partial immobilization to reduce pain induced. Injuries to support lower limbs of dogs that need to be splinted or braced. The Leg Splints are similar to human splints for lower leg, ankle and foot injuries.
Prosthetics allow dogs who have had their limbs amputated (from cancer or other causes) to regain mobility. They can similarly be used for certain congenital defects.
I spent a long time finding a reliable distributor of good orthotics and have the links below. One of the more hard to find items is the drag-bag. This is great for dogs that don't want to use a wheelchair - or for active dogs to use when they are out of their wheelchairs. It prevents raw spots on skin when the dog scoots around and can be fitted with a diaper for incontinent dogs - protecting both the dog and floors.
If you're wondering if a cart will help...here's a wonderful story about a goat that was helped by a cart: Frosty the Snow Goat