Time! You need plenty of time and you need to accept you can’t take everything and some cuttings will fail.
Typically I wouldn’t dig plants up unless you can divide without impacting the garden left behind.
When I was moving I planned well ahead so I could take cuttings at the right time of year. It was very important as some of my plants are “heritage” - they have been handed down by transplant and cuttings from my great-grandmother’s house - so I totally understand how sentimental you feel.
I took lots of cuttings and established them at home first - I gave away the least successful plants on Freecycle and kept the good ones to take to my new home.
Important ones - like roses and a type of lily my late mum loved - I took more cuttings than I thought I’d need so I could allow some to fail in the new garden.
Where possible collect seeds - put them in sealed envelopes and write the plant names on them.
Think about transport. Lots of pots ended up being transported in the boot and footwells of my car! It wasn’t easy.
At the end it was a bit of a scrabble and I did lift parts of some plants that could be easily divided and had gotten overgrown - a type of daisy, some sedums, some rockery plants.
The other thing I did was draw a map of my garden and label it with a key - then I wrote detailed notes about the garden through the seasons, what I knew about the soil, what bulbs were underground and when they’d show, the names of all the main shrubs and perennials, and how to care for some of the more unusual or difficult plants. It felt really good to hand off the garden to the incoming owners with a good chance of figuring out how to keep my totally gorgeous garden alive! I also thought they might appreciate knowing that the rhubarb patch was from original Victorian stock and totally delicious and tender when forced. In reality they probably didn’t care a hoot but it made me feel better!