For mites on caneberries, typically you spray after leaf bud break when new growth is an inch or less, then again a week or two later. This is when over-wintered mites start to crawl about.

An oil spray is already "sticky".

Thanks Larry Gene,
Followup question: The bottle I bought says to use one quart of the oil per five gallons of water. Surely that has to be the rate for a dormant oil spray, right, not for use on living tissue?
I had expected to use a mere five tablespoons per gallon of water. Does anybody know the correct amount to use for blackberries that have already started to leaf out?
Thanks,
ThisBud4U
p.s. one quart per five gallons would be a 1 in 20 dilution. Five tablespoons per gallon would be slightly less than a 1 in 50 dilution.

Answering my own question: found the label at this website:
http://www.stollerusa.com/sites/default/files/pdf/labelinfo13809.pdf

In case anyone needs to know, for caneberries, the dilution they recommend is 1-3 gallons per 100 gallons spray (i.e. a 1-3% solution). Since there are 256 tablespoons per gallon, that would work out to be 2.56-7.68 tablespoons per gallon, so the instructions I got to use 5 tbls/gallon was just about right.

That ratio for pure oil sounds correct. It depends on whether your purchased gallon is already diluted or is 100% oil. The oil is not so much a poison, but a smothering agent for this insect. The PH is fairly neutral and does not harm blackberry foliage. Some people would test a small area first.

Thanks larry gene. yeah, some people meaning smart people. Guess I ain't one of them kind. Just sprayed the heck out of my plants. Guess I'll find out in a week or two whether I burned 'em, and in few months whether I got the critters.

One piece of advice I heard that's worth repeating here for anyone who hasn't heard it: when you're spraying something that kills by smothering, it's best NOT to spray in the heat of the day. Here in Southern California, our days, even in winter, can get upwards of 80 degrees (don't they wish they had winters like that in the Midwest and the East!). So, potentially, Golden Pest Spray Oil could damage the leaves if I were to spray it mid-day. Just a heads up for anyone using this product, or a similar type oil spray. Of course, if you're plant is still dormant, you won't have to worry about burning the leaves 'cause there won't be any!
Hope this helps,
ThisBud4U