Lantana Problems
The damage to these leaves can be caused two different insects, both of which are sucking insects that feed on the leaves. One is whiteflies. These small, flying insects are common in the late summer and both the adult and the nymphs feed on the leaves of a number of perennials, including lantana. The nymphs are also tiny, but look like small, rounded golden orbs. They are mostly on the underside of the leaf. Treatment is very difficult because these insects are mobile, but insecticidal soap is highly effective when used often and before infestations become numerous.
The second is spider mite which are minute insects that are not always visible to the eye. They, too, feed on leaves and when numerous leave damage such as you see and sometimes there are fine webs on the undersides of the leaves. They, are also hard to treat effectively when the infestation is large, but Neem oil is highly effective when the numbers are smaller.
In both cases, lanatana can sustain losing all or most of its leaves to these insects and recover completely if kept well watered.