The poor are much more likely to be in debt, which actually means debt becomes less of a problem as inflation occurs.
The rich tend to have diversified non-cash assets (investments, property, etc) that are immune to inflation.
The middle class -- Joe and Sally with a savings account -- actually have a significant amount of their net worth in cash and that's who suffers.
Mortgage holders are stakeholders in inflation, hoping for appreciating property prices.
Consider the relative size of new mortgages to wages. Since asset prices are typically leveraged on wages (through mortgages), assets will inflate faster than wages (unless the interest rate is set to counteract this.)
But inflation is only beneficial to /new/ buyers if wages inflate faster than assets. Otherwise the relative debt of all mortgage holders increases over time.
Whether asset inflation is beneficial to a home owner is debatable, since all properties appreciate at the same rate. "Moving up" still means increasing the relative size of the mortgage.
This is not strictly true. It's a matter of how money enters the economy. Right now money enters the economy on the supply side through assets which means assets experience inflation. If that money was actually entering through jobs programs or some other demand side mechanism then inflation would happen at the demand side first and on the supply side later.