The Air Nomads have their set of toys, and Korra showed impressive skill in other areas than waterbending. However the case for all of the others would likely be different. So my question is is there any evidence of the other Nation's Avatar selection?
Although the question is interesting, we don't have a reason to believe other nations have a similar way of identifying the Avatar. Air Nomads are likely in this position because:
In general, Air Nomads have a better connection with their spiritual side, making the Air Nomads the most likely canididates for a spiritual test like this.
The Air Nation was the smallest of the four nations and inhabitants lived in secluded places. The other nations would have a more difficult task to screen all newborns in their nation with a similar kind of test.
It is never confirmed or denied the other nations have a way of telling who the next Avatar is going to be. We only know this is the way Aang learned about his powers. Roku was only informed of his powers by the Fire Sages, no reason is given how they know he's the Avatar. While Korra didn't need anyone to tell her, since she figured it out by herself.
We don't know how the other nations handle this issue. Mind you, Korra was identified because her parents saw what she could do and they contacted the Order of the White Lotus. Note that during Aang's tenure they changed from their Freemason ways to a more public group (and we all know what THAT caused) that the common man could somehow contact (note that their leader mentions that there were several duds in the process prior to finding Korra).
However, going by episode 1x02 ("The Avatar Returns") there are several groups all over the world with temples that started to glow when Aang entered the Avatar State. It is quite likely that these groups were tasked with finding the new Avatar in their nation upon the death of the old one, a bit like how the White Lotus found Korra.
I think their nation is identified by the avatar cycle. Water, earth, fire, and air. Aang was air, so Korra is water. But how do we know Aang was air? Roku was fire. But how do we know Roku was fire? Kyoshi was earth. But how do we know Kyoshi is earth? And so on. It all goes to the first avatar, Wan. I'm pretty sure Wan was considered fire nation, and I can support that. The first bending Wan got was firebending. Raava is one of the things that defines an avatar. Before he fused with raava, and became the avatar, he had firebending. After that you can see how the cycle goes.
Hope this helped :)