Will falling dollar cause domestic inflation?

The question:

Since 2002, the U.S. dollar has depreciated over 40 percent against a basket of major currencies, weighted by their countries' trade with the United States. Over the past two years, the trade-weighted dollar has fallen by 15 percent. The decline in the value of the U.S. dollar, particularly if it continues, has raised concerns that it might lead to higher inflation. After all, a lower value of the dollar is likely to raise the cost of imports, which can feed into higher consumer prices.

The answer, from Fed Governor Mishkin:

Sizeable depreciations of the nominal exchange rate exert fairly small effects on consumer prices across a wide set of industrial countries, and these effects have declined over the past two decades. Exchange rate depreciations are thus likely to have less adverse effects on inflation than they have had in the past. The empirical evidence also indicates that pass-through from exchange rates to import prices is low and has declined markedly over the past two decades. This evidence suggests that there may be a weaker relationship between exchange rate fluctuations and nominal demand than prevailed in the past, which may make it easier for monetary policy to stabilize inflation and real activity.

What he seems to be saying is that monetary policy may be the proverbial "free lunch." We can lower interest rates, which makes the US a less attractive place to put money, which leads to dollar depreciation. But a weak dollar will not cause domestic inflation.