@Jferstler: "Arrastrarse", in the sense Robin is using it, means to humiliate yourself; in this case, by sending mail messages to someone who has broken up their relationship with you.
"To fall behind", which is to fail keeping up the pace, in Spanish would be something like "retrasarse"/"atrasarse"/"tardarse" (depending on contexts and regions).
The man is falling behind on sending E-Mails -> Se está retrasando/atrasando/tardando en enviar los E-Mails.
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