AFAIk the DMM7511 and some other Keithly meters use a slightly more complicated AZ mode than the HP meters. It is not just signal and zero, but a more mixed sequence like:
Signal , Zero , Ref , .... or most of the time Signal , zero, but sometimes a different reading (ref., temperature) instead of the zero.
There is some correction for the ADC gain every few seconds - this give the very good warmup drift. However this more complicated mode likely also causes the extra noise in the 1-10 seconds time scale, as the 1/f noise suppression is poor.
These extra spikes could be those cases where an extra reading (like the temperature or the reference) is done.
Line sync mode would add additional delay before the readings. Chances are the spikes may also be visible on a scope connected to the input, when in then 1 or 10 V range.
Many of the other Keithley meters have a buffer at the input, so they would not show such input spikes - the DMM7510 seems to deviate in this aspect and likely has the switching directly at the input, just like most HP meters. It may not have a precharge phase with all the extras like the HP meters to minimize the input spikes.