Can a dishwasher can be used for meat dishes and milk dishes in separate cycles if an empty cycle with soap is run in between?

Shalom, Thank you for your question. There are many varied opinions about using a dishwasher for both meat and milk dishes, even in separate cycles. So, firstly, I would advise you to speak with your local Rabbi, and find out what the local acceptable practice is, and follow that ruling. If, however you don’t have a community, or Rabbi, to follow, then perhaps the following information will be of some help. Rav Moshe Feinstein zt”l ruled that one may use the same dishwasher for separate cycle meat / milk use. However, he required separate racks, and making sure that the machine is cleaned well after each load. Many Rabbis argued with him, or at least, ruled that this ruling only applies that dishwashes that have hot water from an external boiler coming into them (as is common in the USA). But those dishwashers that heat up the water inside the dishwasher itself (such as in Israel) have a stricter law, and can only be used for either meat or milk, but not both. Rav Ovadya Yosef zt”l ruled that the detergent is strong enough to nullify any meat or milk taste. Based on this he allowed the use of a machine for both meat and milk, on the condition that the soap powder was present from the beginning of the hot water cycle. Again, many Rabbis argue with this ruling. Based on these opinions, you can see why some Rabbis do permit using the same dishwasher for (different cycles of) milk and meat. And some require the empty cycle between them, as you mention. And some require separate racks. But, as I mentioned, this is certainly a matter of debate, and you should follow local practice, determined by the local Rabbi. Blessings.