As long as all are silent, I'll add my two cents.
When you are weaving with amplitude, you usually ground yourself - obviusly to prevent falling. It's generally ok in boxing, but such grounding is dangerous when kicks are involved - just because you are cutting your mobility. Also, while laiding down your head, you place it near to your opponent front knee - which usually is charged for quick shield/check-kick and surely can be used for striking your head. Adding weave motion here may lead things go bad.
Also, comparing to hand striking, some kicks have more "attacking surface". For example, when you are throwing a middle kick with front leg (say, as a counter to your opponent's weaving to his back-leg) - it's a side kick - weaving won't help here - because, since middle is usually going from bottom to up, it's just impossible to evade it in such way.
In boxing it is effective, because hand strikes are usually "piercing" or thrown high - you can effectively evade them with bob-and-weave approach. But while kicking is involved, you also involve "non-piercing" kicks - which are generally unaffected by such evading.
In kicks-involved activity it can be used if you are rushing to cut the distance and want to prevent(or, at least, complicate) your opponent from landing a heavy stopping punch. As long as cutting distance effectively lightens damage from most kicks (of course, you still can potentially ate front-kick or ushira-geri to your body), that is the best usage for it here, from my point of view.
Some points about elbows. Generally, it's for very close distance, in fact - clinching distance. Comparing to boxing uppercoats/hooks, elbows have some advance - they are not suffering from clinch distance.
For example, if your opponent is weaving to the front leg - you may (especially if you are taller), ground him with your right elbow, doing side-step with your back-leg. As long as in such upper-to-down elbows you can include your weight, and the distance is very close - result may satisfy you. Also, don't forget down-to-up elbows - also very useful as very-close-distance-uppercoat.
Difference from boxing is huge here. What you may do in boxing, when the opponent is too close? Clinching, and then referree stopping and placing both of you at a distance. In MT/MMA there is no stopping.
All examples above are describing situations, when both of you having left hand as front-hand.
I'm not a grappler, not at all, but maybe you are also more vulnerable to takedowns/throws, if grounded and placing your head low.