Kicks to the knee are very easy to counter if you have an effective hanmi (and not the type very typically seen demonstrated in Aikido); I've most often used nekoashi dachi (collectively called a "Cat stance").
This type of stance allows you to take kicks to the lower legs and still maintain your posture and stance. The only down side is it is very hard to move quickly forwards or backwards in this stance because the majority of your weight is on your rear leg.
On reading the kick though, you always have the option of driving your knee into uke's shins...
Do they take time to plant themselves solidly before they kick?
For someone who is experienced, probably not; Hiza geri and Ashi Barai generally don't have much of a lead in. Knees are very prone to excessive lateral movement and don't take a lot to overextend, so the kick/sweep does not have to contain huge amounts of power.