Great Question.
Just so everyone knows, one could be using 2 different formats when attempting to swap an SD card from one device to another as you've mentioned above. Because of these differences.. you may not be able to transfer images so easily. Sure the image may read that it's a .JPG but can't cross from one system to another based on the difference in formats. Also with swapping like this.. one may corrupt the images or contaminate the SD card during the transfer process.
However, if your camera is wifi/wireless compatible, you can force images to and from one device to another. In other words smart phone have limited capabilities/options when transferring data or images wireless. If you had a laptop/table which has been expanded to suit these types of transfers.. One would then need a power/USB cord to make the connection. Yet if you're allowed access to a wifi/wireless table view your mobile phone.. you can attempt the transfer in this manor.
Lastly, there are SD cards called EyeFi which can be used as an on board wireless/wifi system within your camera to send images to other devices for storage. However these are paid servers and require activation. Otherwise swapping SD card etc.. isn't impossible, but would require exploring both of your devices or even added a few bridge type devices in order to make the transfer successful.
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