Applications built for Raspbian usually do not run on piCore, not because of Kernel but the heavy dependency chain. piCore kernel is based on Raspbian kernel with very few changes required for piCore. Using a custem kernel wouldn't help in your case. The solution is to rebuild it in piCore.
I really have no experience of compiling/debugging this stuff, but the Xserver I compiled on raspbian buster seems to run with no memory problems on picore11 but the X apps have problems with pixmap segfaults. They all only depend on the base C libs(pthread,crypt,z,util,m,c, and ld) raspbian also does some strange kernel vdso stuff for memory handling dependent on arch. I think the problems are due to the code I am compiling and maybe gcc is too strict or not strict enough in checking datatype?
When I copied across raspbian buster base libs into picore image everthing seemed to boot/run fine on a pi4 and a pi zero (using same initrd). I suppose you use seperate initrd to allow people to choose minimal required image but if a general image dist is created it would save repeated initrd image with only /lib/modules/?? being different (I think adding the 4 raspbian /lib/modules/ trees adds maybe 3Mb to initrd but allows any pi to boot from same initrd).
I am guessing I will not even be thinginig about the tinkerboard kernel for 6months, but there could possibly be simple option for adding second initrd mount for the /lib/modules/ kernel dependent (memory handling vdso link into kernel etc) that allow a general pi/ARMcore image for any ARM system.
I don't know if it is worth the extra effort though as superH maybe the prefered future platform for my application?
Anyways any advice.........
Is there a compile/install guide for picore (would 'tce-load -i compiletc squashfs-tools' be sufficient to compile C code and build a tcz?)
I am wondering what method is used/suggested for adding permanent storage and swap space for a picore dev environment.
Would a VM image for a dev environment be available/possible? this would allow everyone that wants to compile a tcz to be working from same environment.