I've had a similar, but not necessarily the same, problem with "e.scan" which is "powered by Kaspersky" (I've posted this today under "Compatability problems"). Not only did my internet slow down, but my system ran at 50% speed (and, like you, it got worse after 20 mins). It seems to have been a compatability problem. After spending two days trying to solve this, I ended up removing the e.scan security suite - disappointing, since I liked it - and reverted back to my original anti-virus (AVG). The only good thing that came out of this was that e.scan couldn't find anything that AVG had missed.
My problem might not be the same as yours, but my suggestion is that you look for anything that might be in conflict with your Kaspersky (and/or e-mail them at their support, if you think that'll help, with a list of your other protections).
It's time someone came up with a freeware utility entitled "ConflictResolver".