I separate the answer into three aspects: conceptual, functional and technical.
A corporate image has ceased to be a static element years ago due to various factors, including the influence of design styles and adaptability to new media.
At the end of the 80s, a style emerged that completely disrupted all the visual arts called deconstructivism, whose main characteristic was to question and challenge all the established fundamental precepts of each discipline. In graphic design, the main action was to deconstruct any dogma concept: static corporatism, readability, balance, Gestalt, among others.
Deconstructivism in Wikipedia
A clear example in corporate image is the change applied to the MTV logo in each screen appearance:
Although several years have passed, deconstructivism left its wake and many of its stylistic bases are still maintained as references. Currently there are some references such as flexible branding, liquid branding, which without being clear design styles, point to the same idea: adaptability.
In its origins, corporate design was only considered for static applications, mainly printed, and on rare occasions applied to media, film and TV or architecture. With the time these media have multiplied so the design had to adapt its presence to them. The increasingly used moving image, 3D design, the internet, and lately applications on devices made corporate design have to adapt. The clearest case is the responsive logo as explained in this answer: variations in design adapted to different sizes and orientations. As for color, the change is given by the different physical and virtual supports to which the corporate image must adapt.
How should I make a logo adapted to different color schemes? Obviously much better if the change is defined as a corporate guideline. The Carrefour supermarket chain presented its new corporate design in 2009 with a considerable number of colors and gradients to be used in its different communication needs (extracted from this answer).
If the corporate design guidelines are already defined, color variations can be added as an annex without the need to modify the logo in terms of shape.
Personally at the corporate level, I would consider a basic range of colors, trying to vary only the hue and relegating changes in luminosity and/or saturation to a second or no change at all (the Carrefour page is only an extreme example to illustrate the answer). A simple practice after making the color variations, is creating a document with all the logos and their colors, changing the document to grayscale, and trying to make the black, dark and intermediate grays suffer only slight variations, the jumping between the grays from one logo to another be the smallest possible. Although this is not an absolute dogma either.
Original logo from vecteezy.com