They can still do this if the old code base is under a permissive license, since permissive licenses like BSD, Apache and MIT are compatible with proprietary licenses.
I think you are confusing re-licensing with sub-licensing, which are not the same. Under copyright law, the copyright holder is granted certain exclusive rights over their work and re-licensing is one of the rights. If the license grants sub-licensing, a licensee can pass on some or all of the rights in the license to a third party. Of the three licenses you mentioned, only the MIT license allows sub-licensing.
The license terms for a sub-license must be consistent with the original license terms, although not necessarily the same. The sub-licensor can use different words as in the original license, but they cannot override the terms and conditions that are required by that license. The sub-licensor cannot sub-license more rights than have been granted by the original license.
Works released under the Apache, BSD, and MIT license can be included in a larger work with a more restrictive license or modifications can be put under such a restrictive license, but the original license must remain intact.
If you are getting your information on re-licensing from the Wikipedia page below, it is wrong.
Mostly true, but only up to an extent: without a CLA, they can't just update LICENSE.txt to replace the contents with the text of the new license and be on their merry way without any trace of the old one.
They can't hide the fact that it was once MIT/BSD/Apache licensed, and they still have to include copies of that original license (and any notices) even after the switch, as that is one of the conditions that contributors make their work available under, and failure to do so would mean the org is in violation if they haven't otherwise received approval.
Even if one has to include the original license and notices with the new version, the original license and notices apply to the portions which were present in the older versions. The new portions added to the software after the license change must be used only as per the new license. MIT, BSD and Apache licenses don't forbid you to use a new license to your own derivative work.
Everything you just wrote is true, but it's not clear why you felt the need to spell it out, especially right here. (No one has said otherwise. No one has said anything that could be mistaken to mean otherwise.)
