Thanks for the inputs that helped me narrow the cause of what I was seeing enough to ask a local Master Gardener and get the following reply:
"The described raining droplets from blackberry leaves in sun are most likely a classic case of guttation, not a sign of disease or pest, and are normal for a well-watered, vigorous plant, especially under fluctuating temps and humidity. Just monitor for overall plant health and don’t worry unless additional stress symptoms appear.
What Causes Guttation?
Guttation occurs when plants expel excess water through specialized structures called hydathodes, typically at the leaf margins or tips, but can also occur on other parts of the leaf surface. This process is driven by root pressure, which develops when soil moisture is high (such as with diligent irrigation), the air is humid, and transpiration is relatively low (often overnight or early morning but can also occur with rapid temperature/humidity changes). In some cases, the droplets are seen “springing” due to the force of expulsion and local microclimate effects on the leaf surface.
Why the Droplets Spring or Arc
While guttation usually presents as droplets forming at the leaf edge or tip, in some plants or under strong root pressure (which is likely during heavy irrigation and hot, sunny conditions), droplets can be forcefully ejected, creating an arcing effect as described. Since the plant is well-watered and temperatures fluctuate, water uptake at the root may exceed what the plant can release by transpiration alone, so guttation pressure pushes droplets visibly outward.
Is This a Problem?
Guttation is generally not harmful and does not indicate disease or pest infestation. It simply means the plant is taking up more water than is leaving via stomata. However, if guttation is extremely frequent or copious, it can indicate overwatering, and long-term, excess mineral residue from the droplets (especially if using hard water or fertilizer) may slightly irritate delicate leaf tissue. Additionally, on rare occasions, guttation droplets can harbor bacteria or fungal spores; if these are reabsorbed, leaf spots can develop, but this is uncommon.
What to Do
• No action is usually needed if the plant appears otherwise healthy and is fruiting well.
• Maintain good soil drainage and monitor irrigation, especially in high temperatures.
• If large amounts of guttation persist, consider reducing watering frequency slightly and check for sticky, white, or crusty residues on leaves (this indicates mineral accumulation)."
I guess I hadn't noticed this before because the droplets are so fine and difficult to see and so probably do happen on other plants. At least I know these plants are getting plenty of water.