Can a UK research council-funded PhD change direction to fall outside the scope of the RC? I am in my first year of a four year PhD programme in the UK. It's through an interdisciplinary doctoral training centre (DTC), and is funded by the EPSRC (who funds a certain number of places at the DTC, who then accepted my application and so granted me the funding). So the funding is administered by my university, but it comes from EPSRC and I am still bound by their funding rules.

This year I am undertaking a number of relatively short research projects, and in September I will submit a project proposal for the remaining three years. It has been stressed many times by my programme directors that the main project must fall within EPSRC's remit, and that the proposal will likely be checked by the EPSRC. For my short projects I have been careful to satisfy this requirement and have made direct reference to EPSRC guidelines in my proposals to ensure that I stay within the remit.

My question is this: suppose I submit a project proposal which gets approved by the EPSRC and I start the project sincerely intending to follow through on my proposal. But over the course of the project various unforeseen issues and distractions the project changes, and moves into an area not covered by my funding research council. Will this cause me problems down the line? Or once I've cleared the hurdle of the project proposal, will I be in the clear, funding-wise?
It is a serious problem if the unforeseen issues make it impossible to fulfill the terms of the grant.  You will need to work with your programme directors about that.  

If the unforeseen issues and distractions merely make you prefer not to fulfill the terms of the grant, than too you could ask your programme directors.  But don't be surprised if they prefer that you do fulfill them.