What problems with the original document, or changes in society, led to later amendments?

I am assuming you are referring to the amendments after
the Bill of Rights. There were many problems and societal forces that required changes.
I will discuss just two.

One set of circumstances that led
to the 14th and 15th Amendments was the aftermath of the Civil War.  Bear in mind that
the Bill of Rights guarantees that the federal government must ensure the rights
contained therein.  There is nothing in the Constitution or the Bill of Rights that
promises the states will preserve any rights at all.  After the Civil War, it became
apparent that the Southern states were going to do everything possible to make life
difficult for the slaves who were now freed.  So the 14th Amendment makes clear in its
language that the states must provide all the "privileges or immunities" of citizenship,
and the 15th makes clear that race may not affect these rights. Without this language,
the states would have been free to do as they pleased.

An
interesting circumstance led to the 22nd Amendment, the administration of Franklin
Delano Roosevelt.  Roosevelt was elected to office four times!  There was a feeling that
this was too long for anyone to be president, so the 22nd imposed term limits on the
presidency.  Roosevelt had died before the ratification in 1951, so he had no
opportunity to be elected a fifth time, but the principle is a good one.  Too many years
of one person at the helm is probably not a good idea.

For
most of the amendments following the Bill of Rights, there is a story. What you might
notice as you examine them is that in many cases they were reactive, rather than
proactive, ways of making adjustments to situations the framers had not contemplated or
responding to changes in society.  Even today, most of the discussion about any new
amendments is quite reactive, for example, the discussion about an amendment to require
marriage be only between a male and female.  This is clearly a reaction to the assertion
of equal rights by gay people, creating a backlash among various groups who believe that
homosexuality is "wrong."