Just like you, your shoes also need conditioning and nourishment to stay healthy.
The natural pair might get dusty and dark with time. Use regular shoe cleaners/polish to retain its original finish.
For a tanned pair, oil the footwear using any regular oil every 2-3 weeks. This will keep its look fresh and shiny and also it will make the leather softer.
Avoid wearing your Desis when it is pouring outside. In case they do get wet, keep them out in the sun to dry and they will be just fine.
When not in use, make sure you keep them away from moisture; safe in the jute bag with the small silica pouch that comes along with your Desis.