When sharing your vision with a multi-echeloned organization, there a several aspects to address beyond simply stating it and reinforcing those tenets with actions.
Since you will have less direct contact with those once or twice removed, you must invest in setting the organizational tone. Complementing the tone is assisting your first line leaders with communicating that vision when opportunities present. Both are deliberate actions that can easily be ignored because of the energy to both do your non-supervisory specific tasks plus directing your first line leaders.
The tone can be shaped indirectly by your leadership style. This is often the most natural - your energy, rate of communication, responsiveness, scrutiny of products and demand for timeliness implicitly set a tone. If following this technique, be very cognizant of how you do the above. This is similar to one's conversational tone - are you loud, cut people off and talk fast? Or are you quiet, allow conversations to flow and are more responsive? Depending on which applies, the respondent will either be comfortable or uncomfortable; react positively or negatively; be inspired or de-motivated. As will be the case for your organization if you default to your "natural syle."
My preference is to be more deliberate. This includes how to talk to the larger audience (those further removed) and using specific terms that convey your intent. Such as the intangible attributes you want each to have or work on, and why (i.e. to create a common/consistent organizational tone). Give examples of what that means so each can relate, and explain to your first line leaders how they can support creating that tone.
Tone setting is one, the other is pointedly calling out instances to exemplify what "adhering to intent" means to those who do not interact with you routinely. So when you have the chance to "make a point" in support of your vision, or complete a distinct task - early on, go with making the point. You can have the first line leader work the distinct action, but the subordinate must appreciate what you are seeking to instill and/or the way you want the organization (including them) to be. Depending on the size and number of echelons, there will be extended time between individual touchpoints, therefore each one must relate to realizing that vision. Also, ensure your first line leaders are giving guidance that aligns with your vision and does not contradict. That aspect must also be actively observed.
The more echelons, the more challenging the task due to scale and time. So have a plan on how to have all of your subordinates receive and realize your vision, because that is the reason you have one.