Using the Python API, it's pretty simple:
from selenium import webdriver
browser = webdriver.Firefox()
print browser.binary.process.pid
# browser.binary.process is a Popen object...
If you're using Chrome, it's a little more complex, you go via a chromedriver process:
c = webdriver.Chrome()
c.service.process # is a Popen instance for the chromedriver process
import psutil
p = psutil.Process(c.service.process.pid)
print p.get_children(recursive=True)
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