Lilacs are a LOT easier to start from a rooted sucker which can be dug from the ground around the mother plant. You could also take one of those branches coming into your yard and pin it to the soil cover lightly keep damp, stake up the end, and in a few months roots will have grown and it can be cut from the mother plant and planted where you like. Al

Maybe you can ask your neighbor if there are 'babies' on his side under the mother bush that you could dig and transplant. Our first spring at our house we noticed about 120 babies under an old half dead bush. I have successfully transplanted most and have given a lot to friends. I dug a hole about 3x the size of the roots so they could spread out and watered them every other day for about 2 weeks. The second year they went from 12" to 2-3' tall and are doing great!