Bubble up or Push down?
Look at Six Sigma as an example. While many firms have embraced Six Sigma, Lean and other similar programs, do we honestly believe that five to ten years ago many CEOs were advocating the value of Six Sigma? Possibly some of them were looking for solutions to reduce cost, waste and inefficiency, but I'm fairly certain few of them were championing "Six Sigma". There were a few, granted, that got on board early, but I suspect that Six Sigma and other similar approaches "bubbled up" - that is, these initiatives and approaches solved problems in a specific team and their use and value spread throughout the organization, and happened to solve problems that the CEO needed to solve as well.
The question becomes then, why did Six Sigma succeed and if it did "bubble up", why was it successful? First, Six Sigma came along at a time when firms were seeking to reduce cost, and Six Sigma gave them a standard methodology and approach. Second, Six Sigma spread through word of mouth, then was adopted as a corporate standard in many cases, rather than the other way around. In other words, it proved its value and had advocates before it was thrust on others. Finally, there is a body of work around Six Sigma, it has a specific meaning and a fairly standard approach, so it can be taught to others.
If all of that is true, can innovation "bubble up"? Or should it be "pushed down"?
If Six Sigma was successful at the lower levels of the organization and was adopted as a management mantra later, that was because for the most part Six Sigma appears to be a tactic or operational effort, while innovation appears to be much more strategic in nature. Incremental innovation is conducted in product teams, but innovation is tightly tied to strategy, and often the management team will need to advocate for disruptive innovation, so it may not bubble up easily. Additionally, innovation has many different meanings, depending on the purpose and perspective of the individual or team. Incremental or disruptive? Product or Service or Business model innovation? Too many different definitions mean that its hard for innovation to be passed from one team to another very easily. Also, since there are different definitions, there are also different approaches. With no standard "approved" approach to innovation, it's hard for one team to pass along what it has done, and what it has learned, to another team unless all the teams agree on an approach.
In many firms, innovation happens in a wide range of product groups, business teams and business functions, but with different intents and different strategies. This means it will be hard for innovation to "bubble up" and management teams should try to create more common philosophies and goals - in other words, sponsor the innovation that is happening and try to organize it, shape it and frame it in ways that create more common approaches. So, innovation should bubble up and be pushed down.