I think these emojis are cute, but I'm not really sure this is a problem that really needs to be solved... Although I'd love a way to specify a set of emojis that I actually want to see on all my platforms. I dunno.
Most implementations of emoji on the web use Apple or Twitter's emoji sets, which are not open source and may put you at risk of copyright infringement. Native implementation is fairly great on mobile at the moment, however desktop is another story, often displayed as just black boxes(most notably in Google chrome).
Some more info on the unicode emoji standard: http://www.unicode.org/faq/emoji_dingbats.html
A bigger problem here is user control because the font stacks on all of the big platforms don't support "mixin" style font use well.
You can illustrate this with an example outside emoji, even: Imagine you're an English speaker occassionally reading documents mixing Latin and Hangul (Korean) alphabets, and the app you're reading the doc in lets you set a preferred font. Most likely that font will only contain Latin glyphs - so where do you get the Hangul from? All of the stacks do support glyph substitution to fetch them from a different font, but users usually have no easy control over what font that will be. On Linux and Windows it works by specifying aliases (i.e. font Y is an alias for font X, so when font X doesn't have a glyph, we go look in font Y instead), but you can only do it in config files and the registry, there's no GUI control (both Linux distros and Windows try to provide some reasonable default aliases) so the user effectively can't pick a font anymore.
The same problem happens with emoji - it's hard for users to pick their "emoji font".