NAD 1983 to NAD 1983 CSRS Transformation
If you check the official definition in the EPSG Geodetic Data Registry, it lists:
Remarks: Used as part of NAD27 to/from WGS 84 transformation for offshore oil operations - see code 8647. Scope: Accuracy 1 to 2 metres. Used for oil industry operations only. Information Source: Various oil industry sources.
The transformation parameter values are zeroes, so it's treating NAD 83 and NAD 83 CSRS as equivalent within 1-2 m of accuracy.
Esri has been working on getting permission to redistribute the provincial NTv2 transformation files but we need to deal with each province separately.
Nova Scotia plans to release (not sure if it's available yet) a new NTv2 file but it converts between ATS77 and CSRS 2010.0. This would replace the older NTv2 file which converts between ATS77 and CSRS 1997.0. The paper is online here.
Even if you got either NTv2 version, there's no transformations available between ATS77 and the "original" NAD 83 that I know about.
Disclosure: I work for Esri and am on the subcommittee that maintains the EPSG registry.
NRCan has a NTv2 web tool that can be used to transform coordinates between the NAD27, ATS77, NAD83(Original), and NAD83(CSRS) reference systems using the binary Grid Shift Files obtainable from the relevant provincial offices. You can use this tool to determine if it's worth going through the effort of tracking down and acquiring the GSB shift files from the provinces for your scenario. (None of the territories have grid shift files yet.)
From this example result for a point in nothern Alberta we can expect a shift of a of 25 to 30cm using NAD83(Original) to NAD83(CSRS) 2002:
+------------------+----------------+---------------+
| | Latitude | Longitude |
+------------------+----------------+---------------+
| Coordinate | 59.53642891 | -118.77687139 |
| Shift (metres) | 0.257 ± 0.172 | 0.288 ± 0.195 |
| Shift (seconds) | 0.008301" | 0.018341" |
+------------------+----------------+---------------+
At time of writing the available shifts in the tool are:
+-----------------------+-----------------+------------------+-----------+
| Coverage | From | To | Grid |
+-----------------------+-----------------+------------------+-----------+
| Canada | NAD27 | NAD83 (Original) | NTV2 |
| Alberta | NAD83(Original) | NAD83(CSRS) 2002 | ABCSRSV4 |
| BC (CRD) | NAD27 | NAD83(CSRS) 1997 | CRD27_00 |
| BC (CRD) | NAD83(Original) | NAD83(CSRS) 1997 | CRD93_00 |
| BC (Vancouver Island) | NAD83(Original) | NAD83(CSRS) 1997 | NVI93_05 |
| British Columbia | NAD27 | NAD83(CSRS) 2002 | BC_27_05 |
| British Columbia | NAD83(Original) | NAD83(CSRS) 2002 | BC_93_05 |
| New Brunswick | ATS77 | NAD83(CSRS) 1997 | NB7783v2 |
| New Brunswick | NAD27 | NAD83(CSRS) 1997 | NB2783v2 |
| Nova Scotia | ATS77 | NAD83 (Original) | GS7783 |
| Nova Scotia | ATS77 | NAD83(CSRS) 2010 | NS778302 |
| Ontario | NAD27 | NAD83(CSRS) 1997 | ON27CSv1 |
| Ontario | NAD27(MAY76) | NAD83(CSRS) 1997 | ON76CSv1 |
| Ontario | NAD83(Original) | NAD83(CSRS) 1997 | ON83CSv1 |
| Ontario (Toronto) | NAD27 | NAD83(CSRS) 1997 | TO27CSv1 |
| Prince Edward Island | ATS77 | NAD83(CSRS) 1997 | PE7783V2 |
| Quebec | NAD27 | NAD83(CSRS) 1997 | NA27SCRS |
| Quebec | NAD27(CGQ77) | NAD83(CSRS) 1997 | CQ77SCRS |
| Quebec | NAD83(Original) | NAD83(CSRS) 1997 | NA83SCRS |
| Saskatchewan | NAD27 | NAD83(CSRS) 1997 | SK27-98 |
| Saskatchewan | NAD83(Original) | NAD83(CSRS) 1997 | SK83-98 |
+-----------------------+-----------------+------------------+-----------+
NRCan doesn't seem to have a corresponding tool to demonstrate the generic nation-wide NAD83(Original) to CSRS conversion results.
This info gleaned from the links in @rgmrtn's answer to NAD83 to NAD83(CSRS) transformation for Northwest Territories, Canada and deserved to be fleshed out for a general not-locale specific answer. Do go vote that one up and read the things I've omitted.