To work laboriously and monotonously: Drudge. (Merriam – Webster)
To walk heavily or slowly: Trudge. (Merriam – Webster)
To proceed slowly or tediously. (Merriam – Webster)
To walk taking slow steps, as if your feet are heavy. (Cambridge Dictionary)
Walk doggedly and slowly with heavy steps. (Oxford Dictionaries)
If someone plods, they walk slowly and heavily. (Collins Dictionary)
If you say that someone plods on or plods along with a job, you mean that the job is taking a long time. (Collins Dictionary)
To walk heavily or move laboriously; trudge: (Dictionary.com)
To proceed in a tediously slow manner: (Dictionary.com)
Plod for Pudsey.
The North West Tonight team are getting on their walking boots and putting their best feet forward for this years BBC Children in Need appeal.