Battery drain current question

Do batteries have consistent drain currents throughout their battery life? So, as a AA battery drains from 1.5 volts down to dead does it stay consistent or does the drain current also drop off as the voltage drops?

Assuming the circuit resistance is constant, then as the battery voltage decreases there will be a decreasing current flow through the circuit according to V=I*R. This is why direct driven lights get dimmer as the batteries deplete.

Different battery chemistries have different discharge curves, but yes, all of them drop current and voltage over time....
As quick and dirty as I can make it....

Alkalines - fall a lot in the first 20% of life (to below nominal typically), fall steadily for 60% of life, fall quickly again the last 20% of life.... bright a little bit, then get dimmer, dimmer, dimmer, too dim, time to find batteries, die
NiMH and LiIon are MUCH flatter curves (better)... fall a little the first 10% of life, then hold nearly steady at just above to just below nominal for 80% of life, then fall quickly the last 10% of life... so "brightest for a few minutes, a little less bright for a long time, then 'is my light getting dimmer?' <rapidly dark> crud where are those replacement batteries.... "
Hope that helps, quick and dirty...
Also, under heavy loads, alkaline curves turn RAPIDLY vertical... so it'd be 30% very rapid fall, 40% slightly less rapid fall, 30% freefall. NiMH and LiIon stay flatter under similar load increases, of course they turn more vertical as well, but not nearly as severe.... 20% "faster fall" 60% steady fall, 20% plummet for way of comparison.
Alkalines (even the fancier "titanium/ultra" marketing alkaline chemistries) are the worst for high-loads, NiCd (2.0+C) and Lithium primaries (up to 2.0, pulse up to 3.0C) are best, and newer LiIon rechargeables and NIMH are in the middle. (1.0C/0.5C respectively) Those are general numbers, there is a wide range, but it gives some idea anyhow of how to compare them (there are at least 3x Li-primary chemistries and several Li-rechargeables, with different C capabilities, so consider this only a rough outline).
**EDIT** looking at Energizer's chart, they even have a NiCD line charted at 5.0C... yikes!

Alkaline AA (2850mAh rated at 0.2C discharge; "E2" is rated at only 2900)page:
http://data.energizer.com/PDFs/E91.pdf

NiMH AA (2500mAh rated, 0.5C discharge)page
http://data.energizer.com/PDFs/nh15-2500.pdf

Li-Primary AA (3000mAh rated, can't find exact C rate (max continuous discharge is 2amps (2/3C), with 3amp pulses at a 2/8 duty cycle..., doesn't fall significantly even at 2C on their charts though) page
http://data.energizer.com/PDFs/l91.pdf
(checkout the "high drain" chart comparing to alkaline)

NiCD page (650mAh, 1.0C discharge rate)
http://data.energizer.com/PDFs/CH15.pdf

Of course, none of them have completely matching charts, but with a little work you can make out the differences