Three weeks ago I bought two rosemary plants from the local nursery. They have both remained in my kitchen, at a window that gets excellent light and they have received adequate water.
Within the past week, I have noticed white specks on the leaves and that the leaves are drying and turning brown. (See photo). The specks were not visible when I bought the plants, but none of the other plants I have in the kitchen have them either (geranium, basil, orchid). I don't think that there was a "migration".
My first problem is that I can't identify the bugs (at least I think they're bugs). And as these plants are intended for food preparation and I am a relative novice, I don't feel comfortable using insecticides without guidance from someone who is more experienced. Botanical Journeys Plant Guides
The white dots could be caused by spider mites which attack plants growing in low humidity conditions.
You can simply rinse the leaves under running water to remove them.
Look at the undersides of the leaves to confirm this diagnosis. If the red mite is the culprit, you will see tiny red dots. They are so minute that you may have to knock them off onto a white sheet of paper to see them.
If the white dots move, they are not the result of spider mite damage but an insect I am unfamiliar with.
If you want to use something stronger than water, neem based insecticides
are safe and effective.
The leaves of your rosemary plants are probably turning brown and falling because the plants have recently been moved from a nursery to your kitchen.
The light level in your kitchen window is almost certainly lower than that light the plants were growing in previously.
Plants eat through their leaves so when less light suddenly becomes available to them, they must drop leaves which are accustomed to high light levels and grow leaves which can process light more efficiently.
They have to do this quickly (before they starve) which is why so many leaves die so suddenly.
Rosemary leaves will also turn brown and die if the plant is growing in soil that is kept too moist.Growing Rosemary