Arpeggios will play in sync but only if you trigger them on time. If you trigger the chord that starts the arpeggiators a quarter note late, for example, the arpeggio will remain a quarter note behind the beat. As in playing in a band, it is your responsibility to play the trigger note on time.

In this manner, arpeggios are different from say a drum machine or sequencer track, that begins when you press a "Play" button. Arps require not just the function being armed (On) you must trigger a key to begin Play.

Not only must the tempo be right, the feel must work. Just because two things are at the same tempo does not ensure they "work" together feel wise.

There is help about changing arpeggio via the Front panel buttons. On the [EDIT] > Select Part > lower "Common" > "Arpeggio" screens at the top you will see "Sync Quantize", this can sync the change between happening immediately, and happening at the next musical value as you set it. This will ensure how the ARP buttons [1]-[8] and/or the SCENE button [1]-[8] (if linked) control changing the arp pattern... You must hit the button to change the current arp prior to the downbeat, this quantize value determines how much before that downbeat.

Arpeggio Sync Quantize (Sync Quantize Value)
Determines the actual timing at which the next Arpeggio playback starts when you trigger it while the Arpeggio of multiple Parts is playing back. When set to “off,” the next Arpeggio starts as soon as you trigger it. The number indicates the clock.
Settings: Off, 60 (32nd note), 80 (16th note triplet), 120 (16th note), 160 (8th note triplet), 240 (8th note), 320 (1/4 note triplet),
480 (1/4 note)

This does not relieve you from the responsibility to trigger the arp on time. You don't really tell us much about your situation (ultimate goal) so we can't make any recommendations as to which should be the master clock. While the keyboard could send clock to your drum machine but you still have the problem of how to best start everything. If you can send the trigger note in via MIDI you may use it to start the arpeggiators.

Thanks for the detailed answer, Phil. This gives me a lot to go on and experiment with.