Apache 2.0 is compatible with GPL 3.0, but how about AGPL? Can I relicense any Apache or MIT licensed software to AGPL?
To my understanding, all licenses that are compatible with GPLv3 are also compatible with LGPLv3 and AGPLv3.
In fact, if you compare the license texts of GPLv3 and AGPLv3, apart from the Preamble, the rest of the license texts are virtually exactly the same, except Section 13. I say "virtually" because of the obvious differences of the names of the licenses, of course.
Section 13 of GPLv3 and AGPLv3 license texts make them compatible with each other for linking, while adding the Remote Network Interaction clause to the AGPLv3.
Hence, the Apache License v2.0 is compatible with the AGPLv3.
However, please note that you can relicense a work based on Apache License v2.0 only when you create a Derivative Work out of it. The original Work would have to be distributed under the original license.
This is based on Section 4 of the Apache License v2.0:
You may add Your own copyright statement to Your modifications and may provide additional or different license terms and conditions for use, reproduction, or distribution of Your modifications, or for any such Derivative Works as a whole, provided Your use, reproduction, and distribution of the Work otherwise complies with the conditions stated in this License.