You mention thickness but you don't mention length or where the cut was on your wood in respect to that length.
Use your fence effectively
From the sounds of it you are cutting near the end of what sounds like short wood. Many mitre saws (single/dual compound) don't have a full fence (so that the blade has room to rotate) and they taper down near the blade. That gives room for wood to move around since it is not supported by the full fence.
The saw will push the wood against the fence so you want to work with that natural tendency. You could make the fence longer by screwing a straight board across the existing fence.
Support the work
If the wood is long be sure that you support the ends because you cut it you will have changed each ends centre of gravity which will make it pop up into the blade.
Hold the work
You can use your hands to hold the wood against the fence provided it is far enough away from the blade. If you are not comfortable holding it as it would put your hand to close then the wood you are cutting is tool small and you are using the wrong tool. Some mitre saw manufactures have clamps that are attached to the fence. Use them if they exist and will fit against your piece. You can use regular clamps as well, again, provided they are away from the operating blade. Clamps are safer to use closer to the blade then your hands. Be careful that the clamp is tightened in case the wood does try and move.
Another thing would be the speed of cutting. If you try and cut too fast the blade could bind and would stop the blade. That sudden force would move the wood. In general you want to be cutting slow enough to give time for the gullets of the blade to remove dust but not too slow that you burn the wood.
Use the right blade for the job
Just like other saws mitre saws have their own blade that are good for cross cutting, rip cutting, plywood, metal etc. Make sure you are using the right blade for your work. If you were cutting down through end grain, for example, with a cross cut blade it will resist and cut very slowly. You risk burning the wood and binding the blade causing damage.
Not sure of the best overall advice but the wood should have as much as possible supported by the full fence. I don't recommend trying to cut would that is not square against the fence either... like a large dowel.