I was told that one of the benefits of doing a convertible note round is that you can do it on a rolling basis?
Issuing convertible notes over a rolling basis is typically only done over a period of a few months (around three months). If you want to keep rolling for a half year or more, you would close one round, and start a new one.
Over a three month period, your valuation typically doesn't change much, so you would offer the same discount/cap throughout the period. If you feel you have a valuation jump, 'close' one round, and start the next one at a new valuation.
Yes, the advantage of a rolling close is funds are available as soon as one investor signs on (unless you agree otherwise with them). I have never seen funds from a rolling close go into escrow.
[Note: spoken from experience as an entrepreneur, not as a lawyer]
For each offering, which usually lasts a few months, you give each investor the same terms. You could theoretically have two or more offerings going on at the same time, and each would have different terms, but I've only seen that once.