The vast majority of the items you might hire a long distance moving and storage company to transport will pose very few problems. If you or the company properly pack up your clothes, furniture, electronics and household items, there's no reason to believe the movers will have trouble with them or anything will be in worse condition than when it left.
However, some items might need special arrangements before they can be prepared to go. For instance, if you have a piano, the logistics of moving it obligate you to let your relocation and moving services know about it well it advance. They might be able to take it, or they might know of a reliable subcontractor that specializes in transporting upright, grand and baby grand pianos.
The same is true for plants. If you're a devoted gardener, you might have a few shrubs or flowering bushes you want to take with you to your new home. However, driving any kind of plant hundreds or thousands of miles is going to put strain on it, so you should ask your moving company if they or their partners can help with the transition. You'll probably want to contact a landscaping business as well to make removing it and fixing the hole it leaves behind less difficult.