Is there any design consideration to keep in mind when switching to higher capacity lithium batteries?
Lithium batteries are crudely divided into two types, 'energy' batteries and 'power' batteries. 

'Energy' batteries are optimised for best energy density, at the expense of power output. They have thin electrodes to allow most of the battery to be used for energy storage chemicals. Due to the higher internal resistance, their output voltage collapses at high loads.

'Power' batteries are optimised for best power output, at the expense of energy storage. They have thick electrodes for low resistance, so less of the battery volume is available for energy storage chemicals.

Power output is often expressed as a 'C' rating. A 60C 1.4Ah battery will output far more current than a 20C 2.2Ah for instance, 84A versus 44A. Check the C ratings of the old and new batteries.