下来/下去
xià lái / xià qù
下来 keeps it in your hand; 下去 lets it run on
The picture Water running downhill. Catch it in your hands and that is 下来; let it pass you and keep flowing and that is 下去. 下来/下去 runs from above downward; 上来/上去 climbs from below upward.
Where the picture shows up in 下来/下去
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down toward me
你从楼上下来一下。
nǐ cóng lóushàng xià lái yí xià
Come down from upstairs for a second.
Coming down the stairs one step at a time from the floor above until you are standing beside me.
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down away from me
我先下去等你。
wǒ xiān xià qù děng nǐ
I'll go down first and wait for you.
A back going down the stairs ahead of me and waiting on the floor below.
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caught and kept
请把地址记下来。
qǐng bǎ dìzhǐ jì xià lái
Please take down the address.
An address that was drifting past your ear running down the pen and settling into letters on the page.
The 下来 in 记下来 is not about downward. It is about holding something so it cannot flow away.
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settling to a stop
车突然停下来了。
chē tūrán tíng xià lái le
The car suddenly came to a stop.
Wheels losing their push, the speed sinking lower and lower until the car settles to a halt.
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flowing on
别停，继续说下去。
bié tíng, jìxù shuō xià qù
Don't stop, keep talking.
A stream of talk that nearly broke off picking up again and running on downhill.
All of it in one picture
下来 and 下去 are one stream running downhill. A person riding that stream toward me is 下来; riding it past me and onward is 下去. Catch a passing address in your palm and put it on paper and you have written it down; let a car's momentum sink until it touches bottom and it has stopped; let a sentence that nearly broke off keep running and you are talking on. So 下来 always carries the feel of catching and holding, and 下去 the feel of letting go and continuing.