"Core the device" - what actually happens

Hi Dick_Dastardly , 
Core files are created for multiple reasons and finding them in your Bigip indicate that something abnormal happened , for instance : 
>> CPU may reach 100 % , so bigip will create core file contains the reasons that made CPU to reach 100 %. 
>> Restarting Bigip services such as ( pccd , mcpd , or bd crashing ..... ) , bigip will create core dump for this.
>> Something caused your bigip to reboot , after opening you will find core dump file. 
>> and other abnormal behaviors can make bigip ip to core. 

So in my opinion , if you troubleshoot during a critical issue related to bigip, you should check first if there are core dump files were created or not. 
>> if you find core file , you must raise a case with F5 support immediately , as Support team is the only direction to help you in reading core files and extract the Root cause for your issue , they equiped sufficiently  to support in this matter. 

>> if you didn't find any core files created , you should proceed in your troubleshooting and see other possibilities that may casued this issue. 

Core files are located in this direction , you can navigate it using CLI " Command line interface " : 

ls -latr /var/core/

Also Take a quick look on this article to know how to manage core files : 
https://my.f5.com/manage/s/article/K10062

I hope my comment helps you 🙂