Global QMS is sensible, rather than complete standalone.
There are usually certain broad 'across the board' things, ie, organisation/group/global requirements that apply to ALL companies in your group. This most commonly includes policies (eg, code of conduct, quality overall, safety overall, environmental overall, HR, etc). It may include the format & content of job descriptions, for example, and for other common elements of business life, eg, performance reviews. It may also include requirements (but not necessarily a procedure) for other areas, including management & storage of records, controlling documents, etc. It might include a common procedure for internal audits, and so on.
The exact number & content of these will vary depending on the scope of your group.
Ideally: decide what is required to be adhered to across the group, and make those your global group requirements.
Then decide what flexibility you want to allow to individual components of the global group in meeting the requirements. You may, for example, enable them to create & maintain many or most of their own procedures, if that works & is efficient. Provided they have clear rules to follow (policies, guidelines) and objectives to achieve (to fulfil the global ones) and there's a good system of management reporting & review in place to ensure it's done, why wouldn't you?
It is important when setting any global/corporate procedures (in particular) to take into account each of your individual companies and their context. I've had to work with local companies (in Australia) who were trying to comply with procedures set globally (eg, USA, Europe) which simply didn't work / make sense here. For example, if global head office has an intranet/extranet and does everything online, whereas some of its component companies don't have personnel with the skills to do this,let alone the IT infrastructure, it won't work.
Make common what works; allow flexibility where it's needed. (Mind you: easy to say, takes a bit of effort to do well!)