Can equipment be enchanted and dis/reenchanted without destroying the item?

If playing a character with a family sword or a particular set of armour that they wish to use throughout the game (from level 1 to level 20), can it start out unenchanted, then undergo a process to gain some kind of Magic Item property, and then much later, be re-enchanted with a more powerful Magic Item property?

I’ve read through the ‘crafting magic items’ section in the DMG but found nothing which mentions whether an existing item can be used for the process, nor whether the item in question can re-undergo the process to change its enchantment.