eyefi brand cards at least, do not dump all the images. Thy can be set to only send images that you select, and thus to only do it when you want to.
I have that set on mine. I don't use the Eyefi for high quality large images, as I only shoot RAW for that.
I use the eyefi to take small jpeg, and I have it set so that it will never auto connect and start sending images.
It only send images when I review them and select the ones I want to send to wifi by using the Canon "Protect" function.
Works great this way. I don't have to worry about excessive battery drain etc.
Another step is that Canon Wifi card enabled bodies have the ability in the menu to disable the wifi function, again another way to prevent needless battery drain.