Whose second? You and I at different points on Earth and our seconds differ. Even a decade ago we could show that a second precisely defined in the middle of the floor of a laboratory could not apply to the whole of the laboratory.

"General relativity states that time speeds up for objects as gravity weakens. To demonstrate this, Chou and his colleagues raised one optical clock 33 centimetres above another. The slightly lower gravity at that height meant that compared with the reference clock, the raised clock ticked with a fractional boost in frequency of 4 × 10^–17, equivalent to a gain of 90 billionths of a second over 79 years."

Even ignoring the distance between different edges of your brain and that there are humans in orbit and in mountain communities as well as below sea level, there are a number of processes in human brains that produce movement an order of magnitude faster (over short distances) not to mention people driving or flying much than the speed in this paragraph:

"To demonstrate special relativity, which says that time slows down for moving objects, the researchers jolted the single atom in their optical clock so that it oscillated at relative speeds of less than 10 metres per second, or 36 kilometres per hour. This time, the clock's ticks seemed to drop by a fractional frequency of almost 6 × 10^–16."

https://www.nature.com/news/2010/100923/full/news.2010.487.h... (Chou et al.'s article is available at http://spiff.rit.edu/classes/phys200/lectures/dilation/chou/...
and starts on the bottom of the second page of the PDF) [and cf my Finally paragraph below]

Importantly when relating an ultraprecise clock on the ground with an ultraprecise clock passing above it in a blimp at, say, a constant linear ten metres per second, a Lorentz transform will need general-relativistic corrections.

These corrections seem negligibly small but it seems safe to bet that in a few years they'll matter in a rack full of high end computers.

Clock precision has also gotten much better http://iopscience.iop.org/article/10.1088/1742-6596/723/1/01... -- "Measurements made with the rubidium fountains (and other clocks at USNO) after the first 1.5 years of continuous operation resulted in the most precise test of the universality of the gravitational redshift and enabled constraints on the coupling of fundamental constants to gravity to be improved by on average a factor of four"

Meanwhile, just to plug some of the work HN's user ISL's group is doing for an extreme setup where the dependence of second not just on position on Earth but on dynamical processes in different stratified-planet layers already matters for practical purposes: https://www.npl.washington.edu/eotwash/node/17 ("seismic Newtonian noise ... the gravitational attraction of an optic to high density regions of the ground caused by passing seismic waves").

Finally, and also because GPS is mentioned here and there in this thread, I recommend slides 16-19 of http://web.stanford.edu/group/scpnt/pnt/PNT18/presentation_f... If we are to insist on the notion of the flow of time, we should be aware that it poses some serious practical problems as one increases the density of marks on a chosen time axis in a dynamical spacetime like ours and populated with sources (in the Einstein Field Equation sense) like those around us. Approaching the limit of ultradense time coordinates we have to consider how parts of ourselves are sources, some organs and regions of organs being denser than others just for starters. And that's true even without getting anywhere near metaphysics and philosophical issues.