I had to solve the same issue and this is what I used as solution.
To use this solution the source and destination table must be identical, and the must have an id unique and autoincrement in first table (so that the same id is never reused).
Lets say table1 and table2 have this structure
|id|field1|field2
You can make those two query :
INSERT INTO table2 SELECT * FROM table1 WHERE
DELETE FROM table1 WHERE table1.id in (SELECT table2.id FROM table2)