I can think of three slightly different “wobbles’ that occur in older voices. Two of them are problems with vibrato. The vibrato either become too slow or it becomes two wide. Slow in the sense of fewer variations in pitch in a given time. Wide in the sense of too much up and down pitch variation. And it is possible that vibrato might have both problems simultaneously, that is, too slow and too wide.
Since vibrato is basically created by the difference in the signals sent by the two major nerves that empower the vocal folds, vibrato problems can occur when age makes the passage of these nerve impulses erratic or slow. This is basically neurological/nerve problem and I know of no one who has explained why it happens to some singers and not others.
A wobble, or what I call a wobble, is usually the result of breathing problems. With age many become more sedentary and this weakens the body core that is the home of the breath muscles, diaphragm for inhale and abdominal muscles for exhale. Since breath management for singing is controlled by the antagonistic action of the diaphragm against the abdominal muscles both of these muscled groups must be kept in good shape. If you exhale a soft, warm, moist breath on your fingertips placed close to your lips, similar to the breath you would use when misting your glasses for cleaning, you will be using your diaphragm to resist the action of abdominal exhaling muscles. It is such a natural action that everyone can do it and we are not aware of how we are doing it.
Yet that warm, moist breath is the perfect breath for singing. Try it. Do the warm moist breath on your finger tips without tone and then add the voice to the warm breathe and you will be surprised how easy it is to produce a lovely tone. And you are doing it by having the diaphragm resist the exhaling action of the abdominal muscles but that is natural
It is very common for singers to be taught some special “support mechanism” to “strengthen” the voice. This usually causes hyper activity of the breathing mechanism; too much resistance of one set of muscles against another set of muscles. The warm, moist breath approach corrects that mistake.
Singers who use any form of hyper activity of the breathing mechanism will likely develop wobble problems, as they get older.