One would be to water them really well, dig them up in the evening with as much soil attached to their roots as possible, put them into the biggest pots you can, water them well again and then leave the pots in the garden in some shade and for a few days afterward. Another way that works for all plant types would need to be done way in advance. Dig a 6′ trench around the base of the plant and go about 12′ to 18′ deep by making a smooth splice through the root system with a sharp shovel and fill this trench with compost. The root ball will then build up strong within a smaller diameter and will be easier to remove. This process is best done 1 to 3 months before the transplant. Plants like monkey grass and liriope can be dug up and the roots cut into sections and replanted right away with little to zero shock. All plants should be checked for damaged stems and removed with clean shears.
When your moving day comes to an end and you arrive at your new home, in the evening would be a good time to plant or keep them in the pots inside until winter is over. Sunlight and good drainage are important as always.
No matter if they adjust in a pot or you place them quickly in the ground the more shade the better during the removal and replanting. Set your plants where you want them and dig a hole twice the diameter and depth and add water and compost and the plant and you are set. Just keep a close eye during hot weather for frequent watering that may be needed. If you have clay soil work in some good soil as you go. Large plants and trees are best in early spring or Late Fall and should have a huge root ball and it’s best to wrap those in burlap to move. The larger the plant the larger the root system. But the planting is easy in a huge hole and water real well.
Transplanting to a near location is a risk and moving them far away to a different climate and soil is even a greater risk. Deciding if the risk and work is worth it, most of the time it is, to move your current garden or favorite plants is up to you. Over the years, people from all countries have moved cuttings and seeds with them — and so can you.