To propagate a variegated dogwood, the most common method is taking softwood cuttings in the summer, where you cut healthy stems with a few leaves, remove the lower leaves, dip the cut end in rooting hormone, and plant them in a moist, well-draining potting mix, keeping them covered with a plastic bag to maintain humidity until roots develop; you can also try taking hardwood cuttings in late fall or early winter, but this method may take longer to root. 

Key steps for propagating variegated dogwood:

Choose the right time: Take softwood cuttings in early summer when the new growth is still pliable. 

Select healthy stems: Choose strong, non-flowering stems with good foliage. 

Cut the cuttings: Make cuts about 6-8 inches long, ensuring each cutting has at least 2-3 nodes (bud points). 

Remove lower leaves: Strip off the bottom leaves from the cutting, leaving only a few leaves at the top. 

Apply rooting hormone: Dip the cut end of the stem in rooting hormone to encourage root development. 

Potting medium: Plant the cuttings in a pot filled with a well-draining potting mix like a mix of peat moss and perlite. 

Provide humidity: Cover the pot with a plastic bag to create a humid environment.

Light and temperature: Place the potted cuttings in a bright, indirect light location with warm temperatures. 

Watering: Keep the soil moist but not soggy. 

Monitor for roots: After a few weeks, gently tug on the cutting to check for root development. 

Important considerations:

Variegation preservation: When taking cuttings, ensure you select stems that exhibit the desired variegation pattern to maintain the unique look of the plant. 

Root development time: Depending on the variety and growing conditions, it may take several weeks to a few months for the cuttings to develop a proper root system. 

Transplanting: Once the roots are established, gradually acclimate the new plants to outdoor conditions and transplant them into the garden in the fall.