<p>A lot of consumer desktop F.D.M. printers comes with a 0.4mm nozzle. I'm looking to print fine details objects and I was considering trying to use a smaller size nozzle. But before I do so I would like to establish a list of downsides and unwanted consequences.</p>

Here are some things to look out for when switching to a smaller nozzle size:
Curling
(out of the nozzle): Make sure the nozzle is clear of any debris to avoid the extruded filament from catching and therefore curling around the nozzle.
Warping
: You might experience more warping on the build plate and delamination between layers as a result of the smaller surface area of the layers.
Reduce speeds
: You should reduce your print speeds anyways when printing fine-detail objects. However, the smaller nozzle size will need a bit more time to adhere to other objects (see above).
Standoff distance
: The distance between the nozzle and build plate, a.k.a standoff, should be a bit smaller with the nozzle size. Typically people use the paper reference (using a piece of paper to "calibrate" the standoff), which is about 0.004".
Make sure your slicing engine knows the change!
Most slicing software will allow you to adjust the nozzle size. This can also be used to fine-tune your machine.
Beware of clogging
: Clogging is usually a result of poor cooling between your heater block and your drive gear, poor filament quality, and/or incorrect extrusion rates. You might want to perform a benchmark print with the new nozzle to "rediscover" which temperatures work best with the new "basin" volume in the nozzle.
I'm sure there are many others, but this should help get you started.