Mifepristone and misoprostol are commonly used for medication abortion in the United States, up to 70 days estimated gestational age. Currently, the WHO recommends giving any hormonal contraception, including progesterone implants or injections, on the day of mifepristone administration.  However, there is a theoretical concern that starting progesterone contraception on the day of mifepristone administration could decrease the success rate of the medication abortion. Mifepristone binds to progesterone receptors; its affinity for the receptor is higher than that of progesterone itself. Administering progesterone in the depot or implant form at the same time could potentially out-compete mifepristone and decrease its efficacy.
A small 2013 study primarily examined satisfaction with the implant on the day of mifepristone; a secondary outcome was a successful abortion. Four out of 20 participants were lost to follow-up, but all 16 who did follow-up had successful medication abortions with no further intervention needed.  Another 2015 study (that has to date only been published in abstracted form) confirms the same outcome. Nearly 500 patients were randomized to receive the implant on the day of mifepristone or after the abortion was confirmed complete. There were no differences in failure of medication abortion between groups. Both were low (< 4%) and consistent with previously published research.  A similar European study confirmed the same, with a non-significant difference (p= 0.47) in abortion success between women who opted for same day etonogestrel implant placement vs delayed placement. 
To date there is only one published study looking at outcomes of depot medroxyprogesterone given on the day of mifepristone for medication abortion. Their primary outcome was again patient satisfaction, but they did evaluate success of abortion. 17 of 20 patients successfully completed their abortions. Two had incomplete abortions and one had a treatment failure; these three underwent surgical evacuation of the uterus. This data suggests against giving depot medroxyprogesterone on the day of the mifepristone; however, it is unclear if the two incomplete abortions would have completed on their own if given longer than the 7-day follow-up period. 
With this limited data, it seems likely that giving progesterone contraception in the form of the current implant on the day of mifepristone for medication abortion will not alter the efficacy of the mifepristone abortion. This should be considered especially in low-resource areas or settings where patients are frequently lost to follow-up. It is probably prudent to wait for larger studies, however, to give depot medroxyprogesterone on the day of medication abortion.
1. Sonalkar S, Hou M, Borgatta L. Administration of the etonogestrel contraceptive implant on the day of mifepristone for medical abortion: a pilot study. Contraception, November 201; Volume 88, Pages 671-673.
3. Barros Pereira I, Carvalho RM, Graca LM. Intra-abortion contraception with etonogestrel subdermal implant. European Journal of Obstetrics & Gynecology and Reproductive Biology, February 2015; Volume 185, pages 33-55
4. Sonalkar S, McClusky J, Hou M, Borgatta L. Administration of depot medroxyprogesterone acetate on the day of mifepristone for medical abortion: a pilot study. Contraception, February 2014; Volume 91, issue 2, Page 174-177.