When riding horses, one of the greatest opportunities for a mishap is when mounting and dismounting. That perilous time when you are neither on the ground nor on the horse's back is often when your horse decides to move or otherwise act up. Thus, it is imperative that all horses learn to stand still for being mounted and dismounted. There are several reasons your horse may not want to stand still for mounting, including anticipation of the upcoming ride, fear/apprehension, physical discomfort, or simple evasion in an attempt to "get out of work," so to speak. We'll address each issue separately.
When attempting to diagnose and correct a horse that wants to move when mounted, you should first rule out any pain/discomfort. Check that your saddle and equipment fits your horse properly, and look for other signs of illness or injury such as lameness. If the riding session is painful, we can surely understand why the horse might be more apprehensive the next time!
Once you have determined that pain is not the issue, rule out fear and apprehension of mounting by going back to basics. Make certain the horse will stand quietly for grooming and saddling. If the horse moves around for these activities, he will likely move when being mounted. Time spent desensitizing the horse to touch and movement will certainly pay off in this case. Recruiting an assistant to hold the horse while you step up into each stirrup, similar to training a youngster to be mounted, can also help acclimate the horse to the mounting process and reduce fear.
For those horses that are anticipating the upcoming ride (usually identified by a need to "prance" or go forward once you are mounted), do not simply mount up and ride away. Instead, take the horse to the riding area, mount, and dismount several times. Then lead the horse back to the grooming/saddling area or stall, and allow the horse to stand tied for a brief period before going back out for your ride or exercise session. When you do mount, don't immediately ride off; instead make the horse stand quietly for a few minutes before beginning your ride or training session. You may have to do this several times before your horse realizes that he doesn't immediately get to "run" or even move forward just because you've mounted up.
Still other horses fidget when being mounted to avoid work. These horses are often identified by the fact that they remain insubordinate once the rider has mounted, such as being resistant to move forward, resisting leg pressure, etc. Mounting up and standing quietly for a few minutes can help (the idea being "this isn't so bad after all") as can mixing up your training routine to include things to challenge your horse mentally and make your ride a bit more exciting for your horse.
In any case, taking the time to teach your horse to stand quietly for both mounting and dismounting is paramount to your safety. If needed, seek the help of a trainer or instructor to assist you in getting your horse to do so.
