The length of the day is increasing roughly linearly at roughly 2 ms (per day) per century, because of tidal effects. The difference between TAI and UT1 is the integral of that, so grows quadratically. If we assume that the mean solar day was "correct" around 1900, then a hypothetical atomic clock that was synchronised to the sun around 1900 is, after x centuries, out by about 36525.x^2 ms. TAI was in fact synchronised to UTC around 1960, when atomic clocks were widely used, and the slowing down of the Earth's rotation is rather unpredictable in the short term (decades), perhaps being affected by climate change, so the numbers are all imprecise, but the long-term (centuries, millennia) quadratic growth of the TAI-UT1 difference is inevitable.
Yes, roughly quadratic increase of LOD over the long term, but over short term more like a random walk. Right now the earth's crust is rotating faster than it did a century ago because things have speeded up. For a view over 2 millennia see plots of LOD at https://www.ucolick.org/~sla/leapsecs/dutc.html