Answer: The glyphosate will kill the entire plant IF it is allowed time to work. The glyphosate kills the plant by being absorbed through the foliage and then translocated down through the plant to the roots -- where it actually kills. It is important to allow the chemical the full time to complete the job; if you pull the plants too soon you will have live root left underground. Read and follow the label directions exactly.
Late August, early September is actually a good time to do this because that is when plants begin drawing down energy and building their root reserves for winter. This means the glyphosate will be absorbed more readily into the root. And, it is a good time to transplant your perennials.
Once you have killed the loosestrife you may want to take to opportunity to add organic matter or do any other soil preparation that might need to be done prior to replanting. This makes the job a little more rewarding.
When you dig the perennials, you may want to bare root them just to be very careful not to reintroduce bits of gooseneck root. Another option would be to take tip cuttings to start new plants and/or easier-to-examine root divisions and replant later using those. You can hold them in containers until replanting time.
Be vigilant and check often in case any shreds of goosestrife remain alive and reappear later. You may be able to stop them in replanted areas by using a wipe on application (rather than spraying) with glyphosate. The wipe on method is easier to control and avoid damaging yoru desired plants.
Good luck with your project!
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