I ended up using a solution that incorporates variables, based on a Stack Overflow answer posted here. This solution appears to be a bit more flexible and efficient than other answers provided.
SELECT u.date,
@running_total := @running_total + u.count AS count
FROM (
SELECT COUNT(*) AS count, DATE_FORMAT(FROM_UNIXTIME(created), '%b %d %Y') AS date
FROM {users}
WHERE created >= :start_time AND created <= :end_time
GROUP BY YEAR(FROM_UNIXTIME(created)), MONTH(FROM_UNIXTIME(created)), DAY(FROM_UNIXTIME(created))
) u
JOIN (
SELECT @running_total := u2.starting_total
FROM (
SELECT COUNT(*) as starting_total
FROM {users}
WHERE created < :start_time
) u2
) initialize;
Note that the group by, date formatting, and range requirements are simply specifics of my particular project. A more generic form of this solution (as per the original question) would be:
SELECT u.date,
@running_total := @running_total + u.count AS count
FROM (
SELECT COUNT(*) AS count, DATE(FROM_UNIXTIME(created)) AS date
FROM {users}
GROUP BY date
) u
JOIN (
SELECT @running_total := 0
) initialize;