  We present a detailed analysis of deep far-infrared observations of the
nearby edge-on star-forming galaxy NGC 4631 obtained with the Herschel Space
Observatory. Our PACS images at 70 and 160 um show a rich complex of filaments
and chimney-like features that extends up to a projected distance of 6 kpc
above the plane of the galaxy. The PACS features often match extraplanar
Halpha, radio-continuum, and soft X-ray features observed in this galaxy,
pointing to a tight disk-halo connection regulated by star formation. On the
other hand, the morphology of the colder dust component detected on larger
scale in the SPIRE 250, 350, and 500 um data matches the extraplanar H~I
streams previously reported in NGC 4631 and suggests a tidal origin. The PACS
70/160 ratios are elevated in the central ~3.0 kpc region above the nucleus of
this galaxy (the "superbubble"). A pixel-by-pixel analysis shows that dust in
this region has a higher temperature and/or an emissivity with a steeper
spectral index (beta > 2) than the dust in the disk, possibly the result of the
harsher environment in the superbubble. Star formation in the disk seems
energetically insufficient to lift the material out of the disk, unless it was
more active in the past or the dust-to-gas ratio in the superbubble region is
higher than the Galactic value. Some of the dust in the halo may also have been
tidally stripped from nearby companions or lifted from the disk by galaxy
interactions.
