Some ceramic capacitors use barium titanate as their dielectric (which is often the case for any larger capacitance values. Smaller values tend to be a different dielectric, one rated as C0G for example).
Barium titanate is ferroelectric, which is very similar to ferromagnetism, but for electric fields rather than magnetic ones.
This famously causes the loss of capacitance of high-value ceramic capacitors under DC bias due to the ferroelectric dielectric saturating as the electric field increases (which increases with voltage) much like the magnetic core of an inductor saturates with increasing magnetic field (which increases with current).
Ferroelectric materials exhibit hysteresis just like ferromagnetic materials. If one is polarized by an external electric field and that field is then subsequently removed, there will be some remnant polarization left in the ferroelectric material. Tiny grains of originally randomly oriented dipole domains have actually been shifted slightly, so the material is now 'magnetized' like a 'permanent magnet', only it is polarized like a ferroelectric dipole analog to a magnet.
This residual polarization must be overcome when reversing the polarity of the field, and at low enough voltages, this is the dominant ferroelectric effect seen. This alignment of the domains stores a little extra energy to reverse that is only seen when you are reversing the polarity of the capacitor (as seen with AC voltage). And of course, once reversed, there is now some remnant polarization in the opposite direction, which must too be overcome.
This doesn't happen instantly, repeated cycles will slowly shift more and more domains off of their initial random arrangement and directed preferentially towards one polarization or another. This is because some will be oriented such that they are more easily aligned by polarization in one direction vs. the other. With the repeated application of the field, the strength of the ferroelectric polarization will slowly grow until tapering off.
Unlike permanent magnets however, this polarization hysteresis is short lived (and the lifetime depends heavily on temperature). The remnant polarization will rapidly be lost when all external field is removed, so if you disconnect the capacitor then start measuring it again, you'll see the whole process start over. But it doesn't vanish faster than the timescales of 1 millisecond like with your LCR's measurement frequency, so it builds up while connected.
Some capacitor manufacturers spec their capacitors at 0.5VAC because this field is too weak to induce meaningful amounts of hysteresis. I suspect if you measure that 68nF capacitor at 0.5VAC instead of 1VAC, you'll see that it will stabilize at 68nF. There is a range slightly above this in the 1-1.5VAC range where you'll see it gain capacitance, and then it will begin to lower again as the voltage increases further and polarization saturation effects start to dominate.
Here is an example of such a curve from Murata. Note the X5R and Y5V curves. Your 68nF capacitor is almost certainly one of these dielectrics or similar.
Regardless, what you are measuring is expected. Capacitors that have relatively high capacitance for their size pretty much always achieve that by employing 'dirty tricks', and these tricks come with odd side-effects and dependences that toss stability out the window.
If your LCR meter supports measuring at 0.5VAC instead though, give it a go!