The first thing you’ll want to do is check to make sure that the cassette on your bike and on the Kickr have the same number of gears, for example if you’re running a 10 speed cassette on the bike, you’ll experience skipping if you’re running an 11 speed cassette on the Kickr, same for the other way around.
If the number of gears match up, it could be that the gear ratios are different. For example if you’re running a 12-28 on your bike and an 11-23 or 25 on the Kickr your derailleur limits won’t be correct and may cause skipping.
You can fix either of these first two issues by buying a new cassette for the kickr that matches the other. Just make sure you have the proper spacers on your kickr to fit a smaller cassette if needed.
The third issue could be due to the age of your bike, if you haven’t been diligent in replacing your chain every few thousand miles, the chain and cassette on your bike could be very worn out, and when you put the old stretched out chain on the new cassette on the Kickr, it may cause it to skip. This is a more complicated issue. If your chain is so worn it has caused the cassette and chainrings to wear with it, replacing the chain alone won’t work, you’ll need to replace the cassette and possibly your chainrings as well. A bike shop can check this easily for you and let you know if this is the case or not and what it will take to get you going.
The fourth is cable stretch, but if this is the issue, you will experience your gears skipping both on and off the trainer. A quick basic tuneup at your local bike shop will fix this.
Hope this helps