Your counter service places that offer limited dessert choices...like the prepackaged dessert cups at Cosmic Ray's in the Magic Kingdom...will travel fairly well (although if you're there when it's really hot, the frosting will melt pretty quickly in the heat.) Most of the non-pre-packaged dessert items will not travel well because they are cold items (in general. There are a few exceptions of course...if cookies are an option, those will travel. At resort counter service you can usually get a big cookie as a dessert. Sometimes fruit if an option, too.)
Our experience is that the vast majority of table-service desserts need refrigeration immediately or are ice-cream based desserts that cannot "travel". Even the desserts that should "travel" for at least a little while--like cheesecake, or layer cakes--at Disney, these often have a whipped cream or creme fraiche layer that will melt within a few minutes. So to be sure, ask your server if the dessert you want will be a melter or not. If it's on the menu, you can choose something like an apple tart or fruit crumble/crisp dessert and ask for it without the sauce & without the ice cream. Those will travel...but our experience on those is if they normally serve them with ice cream or whipped cream: they NEED the ice cream or whip cream. We've thrown out some very dry desserts we took to-go because without the cream they were just too dry to eat. Some restaurants have virtually nothing on the menu that can be a "to go" dessert...for example, we just ate at Via Napoli. Seems like every single dessert was basically some sort of ice cream.