I have a paper on the topic "Accommodating Multiword Expressions in an Arabic LFG Grammar". It's from 2006, but you might find it of some relevance.
> From: agnieszka.patejuk@googlemail.com
> Date: Tue, 26 Aug 2014 15:44:54 +0200
> To: pargram@mailman.uni-konstanz.de; xle@mailman.uni-konstanz.de
> Subject: [XLE] representation of MWEs in LFG?
>
> Hi,
>
> I would like to know if there are any works on representing multiword
> expressions in LFG, taking their internal structure into account?
>
> Example: United States of America
>
> One option would be to treat "United States of America" as one element
> whose PRED is 'United States of America' and which corresponds to one
> leaf at c-structure.
>
> The other option would be to say that "United States of America" has
> internal structure: States as PRED, United as ADJUNCT of States, of
> America also probably as an ADJUNCT of states. The problem is then how
> to represent the PRED for the entire unit (United States of America) –
> PRED-MWE?
>
> Maybe the third option (a variant of the second one above) would be to
> set the PRED of the head of the MWE to the PRED corresponding to the
> entire unit (United States of America) and then ensure, using
> appropriate constraints, that the remaining parts of the MWE are
> present? (^ ADJUNCT $ PRED FN)=c united (^ ADJUNCT $ PRED FN)= of (^
> ADJUNCT $ OBJ)= America
>
> Are there other approaches? I would be grateful for any suggestions.
>
> Best,
> Agnieszka
>
>