Column diameter only affects your sample loading and the flow rate. General rule is if you increase the column diameter twice you have to increase your flow rate four times (square times the ratio of the column diameters) to have the same retention times for your analytes. You also may increase your sample loading on the same rate. If you recall our discussion earlier you have to be careful to keep consistency with your flow cell volume, especially if you are decreasing the column diameter.
If you had decided to go to the very narrow column, which may have sense for the small amounts of sample like an extract from the single cell. Column with the smaller diameter will give you the benefit of the small peak volume and your apparent sensitivity will increase. In this situation you have to use microinjector and microflow-cell. Additional requirement is for your pump, it has to be able to give you stable very low flow rate.
An example. If you are scaling down the method developed for the conventional column (4.6 mm I.D.) with 1 ml/min flow and 10 ml sample injection volume. And you decided to go to 1 mm I.D. column, than you have to decrease your flow down 4.62 = 21.2 times. So your flow has to be 47 ml/min and your injection will be ~0.5 ml. You definitely need special pump and special micro-injector for this analysis.