I've heard that it's best to clean the bore from the breech up on most bolt-action rifles. Does it really matter? I've been cleaning mine the opposite way but I do use a bore guide...
Cleaning from the breach has two advantages, you reduce the risk of damaging the barrels crown and you get less solvent and crud in your action. Damage to the muzzle crown will hurt accuracy. When I am forced to clean from the muzzle, as in some autoloaders and .22 lr, I always use a delrin (type of nylon) bore guide. I'm told the brass bore guides work well also.