SWL does NOT emulate OD&D, with or without its supplements. It emulates Swords & Wizardry. That is its purpose. As such, it doesn't hew exactly to White Box, Core or Complete but works with all of them. You should be able to use spells, magic items and creatures from any of these flavors of Swords & Wizardry with little if any adjustments. Character classes have been greatly simplified with Light, so if there is a hiccup in compatibility, its on the preparation side and not so much in the game session itself.
In the next week or so we'll be putting up a conversion guide for those that want to start with SWL and move on to Swords & Wizardry Complete. While the rules are compatible, characters created using the Light rules are less powerful than those made directly with the Complete rules. We are also putting together some support material to make running current Swords & Wizardry Complete adventures using Light a seamless process.
In addition, +James Spahn and I are working on a player's one-sheet and a GM's one-sheet to add a few classes from Complete (and beyond), leveling to level 5, new spells and for the DM more creatures, magic item guidelines and a bit more. Our goal is for a gaming group to have a complete campaign in the Lost City of Barakus setting using the Light rules. I think we can hit that mark ;)
We treat classes beyond the core four as variants of the core. I'll offer the Bard as an example:
Bards function exactly as thieves, except they may cast Charm Person and Detect Magic once per day. Once per combat they may sing an inspiring song, granting all allies +1 to all attack rolls for 5 rounds. Bards need one additional adventure for each level attained.
As I've said before, Swords & Wizardry Light has legs. Surprising legs.