You can go to larger LUNs with ASM rebalancing and you will ideally want multiple LUNs in the ASM diskgroup to begin with due to either the coarse or fine granularity set at the ASM level. If you just have a single LUN, you won't get any of that striping. As Avi said, it is not uncommon for people to increase their LUN size/LUN layout when going to an XtremIO. If you are choosing ASM as the method of migration, then your process would have you just add these new XtremIO LUNs to your existing diskgroup (alter diskgroup <diskgroup> add disk '/dev/xxx') and then watch the data as it is rebalanced across the new LUNs:
SQL> select GROUP_NUMBER, OPERATION, STATE, POWER, ACTUAL, SOFAR, EST_RATE, EST_MINUTES, EST_WORK, ERROR_CODE from v$asm_operation;
GROUP_NUMBER OPERA STAT POWER ACTUAL SOFAR EST_RATE EST_MINUTES EST_WORK
------------ ----- ---- ---------- ---------- ---------- ---------- ----------- ---------
2 REBAL RUN 1 1 196 1450 0 689
Where SOFAR is how many Allocation Units have been moved
EST_WORK is the number of Allocation Units left for movement
EST_RATE is the number of Allocation Units/Miniute that are being moved.
This will take a while depending on how much data you have and how many LUNs you are working with since it is a host/DB-based migration method. Once it is all rebalanced, you can then drop the original source volumes (alter diskgroup <diskgroup> drop disk <asm_diskname>). This will kick off another rebalance at this point to move all the of the extents from the disks you are dropping to the rest in the diskgroup. I would suggest doing this one disk at a time. There are a bunch of other parameters you can tweak during an ASM rebalancing and as you can see it could take much longer than a one-time SAN or LVM migration, but should give you the desired outcome you seek.
Here is an overview that might help: Administering ASM Disk Groups
I would also suggest talking over the game plan with your DBA as they might have some experience or tweaks they can assist with (I'm not a DBA....I just play one on TV and have tested this in labs )