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- Performance-wise, I'd benchmark this vs `if(memchr(src,'\0',n)==src+n) memcpy(dst, src, n);` because it isn't obvious at least to me if that's faster or slower than your custom function.
- Regarding where `end` is pointing, I think that's the right call since it's convenient to have a pointer to the null terminator for many reasons, such as when determining the string length or indeed some manner of "chaining".
- Bug: no const correctness. Okay so it isn't an actual bug, but it's something with overwhelming consensus of being good practice. In this case it means that you should do `const char* src`.
- Minor: No restrict correctness. As written,`dst`/`end` and `src` may _not_ overlap (and you need to document this). Therefore, `src` can be `const char* restrict`. You can't restrict `dst` and `end` though since if used correctly, they point at the same buffer. 
- Aligned word copy improvements are possible. If you truly wish to write a library-quality function, it would do the copying on word-basis. This means handling misalignment at the beginning, then making an assumption of how far word copying can be done without reading out-of-bounds. Not exactly trivial - you can look at the Github project for glibc implementation of `strcpy` for inspiration. Note that glibc might rely on non-standard extensions however.
- Style remark: `if (!*dst)` looks a bit weird and there's no obvious reason why you wouldn't write `if (*dst != '\0')`.