Are you planning to do this yourself?

> Can anyone explain to me the process/procedure for balancing the
>  internal parts of an engine....pistons, rods, etc, and dynamic balancing
>  of the entire rotating assembly.....pistons, rods, crank, flywheel and
>  clutch plate. How do they do that on automotive engines? Do machine
>  shops literally grind a little material off the piston skirts or some
>  other place until the pistons match the lightest one in the bunch?

Yes

>Is it similar for rods. Where is the typical location to remove material?

Yes, weigh the big end, them little end matching weight to lightest, then
over all weight.

What
>  considerations must be given in each case? How would one balance a
>  crankshaft?

It is put in a "lathe" type machine where it is spun to @500 RPMs then the
computer tell the machinist where and how much to remove from each journel
along the length of the crank.  Then the flywheel is attached and it is spun
again and balanced.  Then the hub is attached and the process is done again,
then the pressure plate is attached, marked and then spun again to balance.

IMHO This might best left to the shop with the correct tools.

> And where would the metal be removed?

Along the crank journels where the computer tells the machinist to remove.

> And what do you look out for?

Removing too much...

Demello will balance the engine and rebuild the rods for @$150 100 balance
and 48 to rebuild rods.

The basic process for rebuilding the rods is:  The little end is rebushed,
then a small amount of the end cap is ground off then bolted to the rod then
the center is set and rebored along the center line of the "corrected " true
center line, then they are balanced end for end and total weight.

Balancing "ALL rotating parts" as a unit will give a smoother running engine
because balanced parts bolted together will still be slightly unbalanced.

Robert