In general Round-Robin DNS works well with static web sites. Once dynamic information is utilized then there may be problems. The problem is that a users browser starts talking with one server and then "randomly" flips to another server. The 2nd server has not idea about the first one and what data was already provided.
An example of a problem with webNetwork is if the user is using Image Authentication. The users browser asks the DNS server for the address(s) of the webNetwork server. It is give back one or more IP addresses. The browser then decides which one it will connect to. It starts talking to Server A and then asks for the images for Image Authentication. If the browser times out the cache for the DNS name of webNetwork it will choose a different IP number in the list and try connecting to that. So it then asks Server B for some of the images for the Image Authentication and those will fail because the names of the images are random on each relay for security reasons.
Another problem with DNS Round-Robin is that if one server is down , users will still try to connect to that server and will get random errors. DNS Round-Robin does not detect if a server is not working. Only full load balancers have that feature.
Another type of webApplication that has problems with Round-Robin is TCP Proxy webapplications. The user clicks on a link and the Relay A starts to set up a tunnel between the user and the back end server. If when the client (for example RDP an RDP Client) makes the connection it is sent to Relay B then that server does not have the tunnel configured and thus the application won't work.