Putting BPA free plastic in the dishwasher is generally safe as long as you place the BPA free plastic or any other plastic dishes on the top rack of the dishwasher which keeps it away from the heating element that could melt the plastic.
Just make sure the BPA free plastic dishes you are washing are labeled dishwasher safe because some forms of plastic including BPA free plastic may not be safe in the dishwasher.
I've washed a lot of BPA free and other plastic dishes in my dishwasher on the top rack though and never has it melted or been a problem.
It really becomes a problem when you place the plastic dishes on the bottom rack due to the heating element heating the plastic up too much and potentially melting it.
But on the top rack the BPA plastic dishes should be just fine as well as many other plastic dishes.