"global grassroots" is correct and usable in written English.
You can use it to refer to a global movement of grassroots organizations or initiatives, often characterized by being organized from the local level upwards. For example, "The global grassroots movement to combat climate change is gaining traction."
Desperately seeking tenure, he is planning a book on global grassroots activists.
On Monday, the Guardian reported an initiative by Google, Facebook and Twitter to foster a global grassroots counter-speech movement.
* * * More than 1,400 charities and businesses are joining the global grassroots movement to give something back on #GivingTuesday tomorrow 1 December.
McKibben has been an outspoken activist against the Keystone XL pipeline, and in 2008 he co-founded 350.org, a "global grassroots climate movement".
The ONE Campaign, a global grassroots organisation of six million members, seeks to hold governments to their word on the issue it sees as critical to ensure responsible extraction – transparency in the dealings of energy and mining companies.
Now, in his new role as CEO of Circular Economy, a Netherlands-based social enterprise, Ridley wants to build a similar global grassroots movement to accelerate the circular economy.
and working for peace and justice through their global and grassroots Peace Corps-like Mennonite Central Committee, as well as humanitarian efforts via the Mennonite Disaster Service and international development efforts supplied by MEDA, an NGO with whom I met when I was last in Afghanistan.
Grassroots Global Governance not only explains why some efforts succeed and others fail, but also why the process of implementing global ideas locally causes these ideas to evolve.
This makes them grassroots global governors.
In addition, Natalia is the founder of Awesome Foundation MIAMI, the local chapter of a global network of grassroots micro-grant-giving hubs that, cumulatively, have awarded more than $1.6 million globally to small, local ideas.
Stop by and become part of this grassroots global food community!
