what do you (as examiners and/or sensei) usually expect of early kyus? (4th-2nd kyu)
It really all depends. [Which you have seen.]
At 3rd Kyu you already are required to do Koshinage.
[while second nature to some Uke Monkeys flying over peoples backs, to others it can be less comfortable, unless they 'roll' around.]
Also at this stage, if not sooner, most people are trying the high flying ukemi for Kotegaishi.
At 2nd Kyu it seems there is a ton of Koshinage. [So you better love Ukemi]
To throw in there Sumi Otoshi [corner drop], with the high flying Ukemi, and Kubi Nagi [Neck Throw... something you definitely want to be careful with.]
Personally I like to keep my feet on the ground as much as possible.
I feel more comfortable with BJJ than I would with Judo.
At the end of the day, even within a dojo, people are going to vary within a belt range.
Unless you take Yoshinkan Aikido. [not to generalize]
We have 6 kyus, and for them its pretty much ok to be more relaxed and basically know the name of the movement and how it looks.
At 3rd Kyu I started to try to figure out what worked and didnt and why it did or didnt.
What was for 'show' and what is practical. [Did this by doing some live training and finding things out the hardway - like getting my lights punched out in Thai Boxing, and pulling off kotegaeshi in the grappling portion... but definitely not the way you would train with it in class.]
Still havent resolved all of this, but Im more convinced that BJJ, Aikido, Judo would benefit [me at least] as one whole art due to the fact that they deal with various ranges/distances.
Enough of my theories though. [Took about a two year break and just started back the beginning of this year so Ill test for 2nd kyu, hopefully, this summer... if I can learn to love flipping over peoples backs more.]
At the end of the day, have fun...